ELECTROMAGNETIC BOUND STATES IN THE RADIATION CONTINUUM ANDSECOND HARMONIC GENERATION IN DOUBLE ARRAYS OF PERIODIC

DIELECTRIC STRUCTURES

By

REMY FRIENDS NDANGALI

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2011

c⃝ 2011 Remy Friends Ndangali

2

My family’s journey is a miracle we owe to the kindness of a multitude of friends, and thecharity of perfect strangers. To all of you who showed us the way in our days of despair,

I dedicate this work.

3

ACKNOWLEDGMENTS

I would like to thank my doctoral advisor Dr. Sergei Shabanov for a wonderful

adventure over the course of my Ph.D program. Thanks for the many lessons in

mathematics, physics, and life in general. Thanks for the encouragements in the

moments of hardships, and the celebrations on achieving milestones.

I would also like to thank my other doctoral committee members, namely, Dr.

Klauder, Dr. Pilyugin, Dr. Gopalakrishnan, Dr. Tanner, and Dr. Hebard, for their interest,

questions and suggested improvements to this work. I would like to thank especially Dr.

Klauder for his lessons on mathematical methods of theoretical physics.

I would also like to thank all the other professors whose classes I took during the

course of the six years I spent in the Ph.D program at the University of Florida. Special

mention should be made of Dr. Block and Dr. Brooks for teaching me Analysis, and

Dr. Turull for teaching me Abstract Algebra. I am also very grateful for the advice, and

teaching of Dr. Robinson throughout the years.

Last but not least, I would like to thank the staff at the Department of Mathematics

for all the help and advice throughout the years. I thank especially Gretchen, Sandy, and

Margaret for keeping me in check, and making sure I never overlook a deadline, or a

requirement. I thank also Connie and Marie, for their availability to assist at all times.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Historical Context: Bound States and Siegert States in the Theory of

Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 The Extension to Electromagnetism and its Challenges . . . . . . . . . . 171.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 ELECTROMAGNETIC SIEGERT STATES IN PERIODIC STRUCTURES . . . 22

2.1 Electromagnetic Siegert States . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Regular Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Perturbation of Bound States in the Radiation Continuum . . . . . 312.2.2 Near Field Amplification Mechanism . . . . . . . . . . . . . . . . . 35

2.3 Decay of Electromagnetic Siegert States . . . . . . . . . . . . . . . . . . 402.4 Proof of the Regular Perturbation Theorem . . . . . . . . . . . . . . . . . 44

3 ELECTROMAGNETIC BOUND STATES IN THE RADIATION CONTINUUMFOR PERIODIC DOUBLE ARRAYS OF SUBWAVELENGTH DIELECTRICCYLINDERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 Scattering Theory and Classification of the Fields . . . . . . . . . . . . . . 503.2 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Bound States Below the Radiation Continuum . . . . . . . . . . . . 603.2.2 Bound States in the Radiation Continuum I: One Open Diffraction

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.3 Application: Zero Width Resonances and Near Field Amplification 67

3.3 Bound States in the Radiation Continuum N, N≥ 2 . . . . . . . . . . . . . 723.3.1 Bound States in the Radiation Continuum II: Two Open Diffraction

Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 A RESONANT GENERATION OF SECOND HARMONICS IN DOUBLE ARRAYSOF SUBWAVELENGTH DIELECTRIC CYLINDERS . . . . . . . . . . . . . . . 86

4.1 The Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2 Subwavelength Cylinders Approximation . . . . . . . . . . . . . . . . . . . 93

5

4.3 Amplitudes of the Fundamental and Second Harmonics . . . . . . . . . . 944.4 Flux Analysis: The Conversion Efficiency . . . . . . . . . . . . . . . . . . 99

APPENDIX

A COMPLEMENTS I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A.1 The Lippmann-Schwinger Integral Equation . . . . . . . . . . . . . . . . . 106A.2 Solution of the Lippmann-Schwinger Integral Equation in the Zero Radius

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107A.3 Complements on Bound States in the Continuums I and II . . . . . . . . . 111A.4 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B COMPLEMENTS II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B.1 Estimation of ζ and ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116B.2 Complements on the Flux Analysis: Flux Conservation . . . . . . . . . . . 117B.3 Complements on the Amplitude E1 . . . . . . . . . . . . . . . . . . . . . . 120

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6

LIST OF TABLES

Table page

3-1 Existence of solutions to systems (3–45) . . . . . . . . . . . . . . . . . . . . . . 81

7

LIST OF FIGURES

Figure page

2-1 Examples of structures considered and the Ckx -plane. . . . . . . . . . . . . . . 24

2-2 The ξ-plane for the integral in Eq.(2–25) . . . . . . . . . . . . . . . . . . . . . . 44

3-1 Periodic double arrays of dielectric cylinders, and the corresponding spectrumrepresentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3-2 Values of the parameters a, kx , R, and εc susceptible to allow the formation ofa bound state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3-3 Plots of the amplitudes of bound states in the radiation continnum . . . . . . . 68

3-4 The specular coefficient and the electric field on the cylinders near a boundstate in the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4-1 The scattering process for normal incident radiation, and the correspondingbound states in the radiation continuum . . . . . . . . . . . . . . . . . . . . . . 87

4-2 The conversion efficiency and its region of validity . . . . . . . . . . . . . . . . 103

8

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

ELECTROMAGNETIC BOUND STATES IN THE RADIATION CONTINUUM ANDSECOND HARMONIC GENERATION IN DOUBLE ARRAYS OF PERIODIC

DIELECTRIC STRUCTURES

By

Remy Friends Ndangali

August 2011

Chair: Sergei V. ShabanovMajor: Mathematics

Electromagnetic bound states in the radiation continuum are studied for periodic

double arrays of subwavelength dielectric cylinders. These states are similar to the

bound states in the radiation continuum in quantum mechanics discovered by von

Neumann and Wigner. For the system studied, these states are shown to exist

at specific distances between the arrays in the spectral region where one or more

diffraction channels are open. The near field and scattering resonances of the structure

are investigated when the distance between the arrays varies in a neighborhood of its

critical values at which the bound states are formed. In particular, it is shown that the

near field in the scattering process becomes significantly amplified in specific regions of

the array as the distance approaches its critical values. This amplification is explained

through the Siegert state formalism, which is extended from quantum mechanics to

Maxwell’s theory of electromagnetism. The said amplification is also used to control

the second harmonic generation in a periodic double array of subwavelength dielectric

cylinders with a second order nonlinear susceptibility. The conversion efficiency of the

incident fundamental flux into the second harmonic flux is shown to be as high as 40%

at a distance between the arrays as low as half of the incident radiation wavelength.

9

CHAPTER 1INTRODUCTION

1.1 Motivation

This work is devoted to the study of a special class of solutions to Maxwell’s

equations that describe the resonant scattering of light on periodically structured

materials and possible applications to optical nonlinear effects in such structures.

In the asymptotic region away from the scattering structure, a solution of the wave

equation can always be represented as a superposition of plane waves characterized by

a specific set of (spectral) parameters, namely, the wave numbers. Spectral parameters

describing the asymptotic (or scattered) waves form a continuous set called the radiation

continuum. Solutions in the radiation continuum are not localized in space, i.e., they are

not square integrable.

For some structures, Maxwell’s equations also admit localized solutions which

are square integrable (e.g., a standing wave in a metal cavity or propagating waves in

defects of a photonic crystal). However, the spectral parameters of a localized solution

are not usually in the radiation continuum (this is the case for the aforementioned wave

guiding modes in metal cavities or defects in photonic crystals).

There is a similar classification of solutions of the Schrodinger equation that

describe quantum systems. Localized (square integrable) solutions of the Schrodinger

equation are called bound states. Consequently, if spectral parameters of a bound

state lie in the radiation continuum, then it is called a bound state in the radiation

continuum. Such bound states were first predicted to exist in 1929 by von Neumann and

Wigner [1, 2], and they are known to be rather rare in quantum systems.

The present study is devoted to bound states in the radiation continuum in

Maxwell’s theory of electromagnetism and their applications. It will be shown that, in

contrast to quantum mechanics, there is a simple principle for designing scattering

structures that support electromagnetic bound states in the radiation continuum. The

10

structures that have bound states in the radiation continuum exhibit unusual scattering

properties which can be used in practical applications to nanophotonics. One of the

most important applications is the development of a novel mechanism to enhance and

control optically nonlinear effects in nanophotonic devices, the problem that should be

solved in one way or another in order to achieve an ultimate goal of nanophotonics: all

optical data processing.

To be specific, in the so called Transverse Magnetic Polarization (TM polarization),

Maxwell’s equations reduce to the scalar partial differential equation,

1

c2∂2tD = �E (1–1)

where c is the speed of light, and D is a field that depends analytically on the field E ,

i.e.,

D = εE + χ2E2 + χ3E

3 + ... (1–2)

Here, ε is called the dielectric constant, while the χn’s, n = 1, 2, ... are higher order

nonlinear dielectric susceptibilities. In the vacuum, E = D. The regions of space in

which E = D represent the scattering structure. In the spatially asymptotic region,

E = D and solutions to Eq. (1–1) are linear combination of plane waves characterized

by their wave numbers that form the radiation continuum. In practice, the nonlinear

susceptibilities are very small in comparison to the dielectric constant ε, and as such,

they are usually neglected, leading to the well known linear wave equation,

ε

c2∂2t E = �E (1–3)

An electromagnetic bound state is a square integrable solution to Eq. (1–3). Consequently,

if the wave number of a bound state lies in the radiation continuum of the scattering

system, the said state is an electromagnetic bound state in the radiation continuum.

Clearly, such states exist only for a special choice of the function ε.

11

It appears that if the scattering system admits bound states in the radiation

continuum, then when illuminated by an incident radiation, the field E may be significantly

amplified in some subsets of support of the function ε− 1 as compared to the amplitude

of the incident radiation. According to Eq. (1–2), nonlinear susceptibilities can no

longer be neglected, which leads to an enhancement of optically nonlinear effects

in the scattering structure. A complete theory of these effects require solving a full

nonlinear scattering problem for Eq. (1–1). It will be shown that when a bound state

in the radiation continuum is present in the system, conventional methods of solving

this nonlinear scattering problem fail because of the nonanalyticity of the solution in χn.

One of the goals of the present study is the development of a rigorous mathematical

formalism to circumvent this difficulty and apply this formalism to some practical

problems such as higher harmonics generation.

However, before we may proceed to give an overview of the proposed novelties,

it is worthwhile to give a simple example from quantum mechanics that clarifies the

significance of bound states in the radiation continuum as well as to stress the key

mathematical differences in the very definition of quantum and electromagnetic bound

states in the radiation continuum. The latter differences comprise a main mathematical

problem that has been solved in the present study to extend the concept of bound states

in the radiation continuum to electromagnetism.

1.2 Historical Context: Bound States and Siegert States in the Theory ofQuantum Mechanics

Using suitable reduced units, consider the single particle Schrodinger equation,

i∂

∂t(r, t) = H(r, t) (1–4)

with hamiltonian,

H = −1

2� + V

12

The potential V is a real valued function that we will assume to be radially symmetric

in the infinite three dimensional space, i.e., V (r) = V (r). It is customary to seek the

solution to the Schrodinger equation as a superposition of harmonically time dependent

states (r , t) = E(r)e−iEt , where E is the energy of the particle. The amplitudes E

are then eigenstates of the hamiltonian H. Specifically,

HE(r) = EE(r) (1–5)

When the potential V is identically zero, i.e., for a free particle, the eigenstate E takes

the form [2, 3],

E,f (r) =sin(kr)

kr, k =

√2E

where it is understood that if E < 0, the square root with positive imaginary part is

taken. The choice of the particular solution E,f is made to ensure that the eigenstate

E remains finite throughout space, and therefore at the spatial origin in particular. The

requirement for the free state to remain bounded at the spatial infinity (r → ∞) implies

that k > 0. The states E,f thus defined are the free states of the particle.

If V is not identically zero, but V (r) = 0 for all r > r0 for some r0, then the quantum

scattering theory requires that for r > r0, each eigenstate of the hamiltonian H be a

superposition of a free state, and an outgoing spherical wave r−1e ikr that occurs as a

result of the scattering of the free state by the potential V [2, 3]. That is,

E(r) = Isin(kr)

kr+ S

e ikr

r, k =

√2E , r > r0 (1–6)

The amplitude S is then called the scattered amplitude of the state E , whereas I

designates the amplitude of the free state.

For positive energies E , the state E remains bounded throughout space. It remains

oscillatory in the spatial infinity(r → ∞), and therefore, such a state is not square

integrable on [0,∞). It is called a scattering state.

13

For negative energies, the number k =√2E is pure imaginary with positive

imaginary part, and therefore the eigenstate E is bounded in space if and only if I = 0.

It follows in particular that if such a state exists, it is square integrable on [0,∞) since it

decays exponentially in the spatial infinity. Such a state is called a bound state.

Thus the spectrum of the energies E of the hamiltonian H is composed of the

scattering states which exist for E > 0, and the bound states that exist for E < 0. An

important remark is that the scattering states do exist for all E > 0. In particular, their

spectrum is the continuum (0,∞), which in the jargon of quantum mechanics is referred

to as the radiation continuum. On the other hand, the spectrum of the bound states

is necessarily a discrete subset of the interval (−∞, 0). This is because the problem

of finding the bound states reduces to an eigenvalue problem for the hamiltonian H

on the Hilbert space L2([0,∞)). As the said hamiltonian is self-adjoint, the spectrum

of the bound states must necessarily be discrete. Since for the bound states just

described, the energy E is negative, these states are referred to as bound states below

the radiation continuum.

The remarkable observation made by von Neumann and Wigner is that if the

condition on the potential V to vanish for all r > r0 is relaxed, and replaced by the

more general condition V (r) → 0, as r → ∞, then some bound states may be

found in the radiation continuum of the scattering states. This is a peculiar, and rather

rare feature in physical systems. In fact, in their paper [1], von Neumann and Wigner

used a constructive approach to derive a potential V providing a bound state in the

radiation continuum, and the so obtained example was rather artificial and of no physical

significance. This led to bound states in the radiation continuum being considered as

mathematical curiosities until a physical example was found in atomic physics much

later[4]. For the curious reader, the actual potential produced by von Neumann and

14

Wigner is1 ,

V (r) = − 64k2A2 sin4(kr)

[A2 + (2kr − sin(2kr))2]2

+48k2 sin4(kr)− 8k2(2kr − sin(2kr)) sin(2kr)

A2 + (2kr − sin(2kr))2, k > 0

For A a nonzero arbitrary constant, this potential produces a bound state of energy

E = 12k2 > 0, and therefore this state is indeed a bound state in the radiation continuum.

Its explicit expression is [2],

(r) =sin(kr)

kr (A2 + (2kr − sin(2kr))2)

The significance of bound states in the radiation continuum is understood through

the theory of scattering resonances. Resonances are peaks (maxima) of the so called

scattering cross section, σ, which is simply the probability flux per unit solid angle, per

unit incident flux. It’s expression for an eigenstate E is,

σ(E) = 4π

∣∣∣∣SI∣∣∣∣2

Peaks in the scattering cross section are explained through the Breit-Wigner theory [5]

by the presence of complex poles in the scattering cross section when it is extended,

as a function of the energy, to the complex plane. Each such pole is called a resonance

pole. It is simple, and it lies in the lower half of the complex plane, i.e., a pole is of the

form En − i�n, �n > 0.

In the vicinity of a resonance pole, the scattering cross section has a Lorentzian

profile of resonance width � described by the well known Breit-Wigner formula:

σ(E) ∝ �2n(E − En)2 + �2n

1 The original paper [1] by von Neumann and Wigner contained a minor error. Theformula produced here is the corrected formula as provided in [2].

15

In 1939, A. J. F. Siegert [3] showed that in general, poles of the scattering cross

section correspond to eigenfunctions of the hamiltonian H with specific complex

boundary conditions. In the case of a potential V that vanishes for all r > r0 as

described higher, the Siegert states are defined as eigenfunctions E of the hamiltonian

H satisfying the boundary condition,

∂

∂r(rE)− ikrE

∣∣∣r=r0

= 0, k =√2E (1–7a)

This condition amounts to requiring that the incident amplitude I in Eq.(1–6) be zero. In

particular, for this specific case, bound states below the radiation continuum are Siegert

states with real negative eigenvalues En.

For a general spherical potential vanishing at the spatial infinity, the boundary

condition for defining the Siegert states becomes,

∂

∂r(rE)− ikrE

∣∣∣r=r0, r0→∞

= 0, (1–7b)

Just as in the previous case, a Siegert state n corresponds to a resonance pole

En − i�n. When such a pole is real, i.e., �n = 0, then the Siegert state becomes a bound

state below or in the radiation continuum depending on whether En < 0 or En > 0.

The physical significance of Siegert states with �n > 0 can be understood through

the initial value problem for the time dependent Schrodinger equation (1–4) in which a

wave packet i , initially positioned in the asymptotic region r > r0 (i.e., at the initial time

t = 0 the support of i lies in the asymptotic region), propagates into the scattering

region r < r0 and passes through it giving rise to scattered waves. In the time-dependent

picture, the amplitude of each Siegert state (with a sufficiently small �n) that has

been excited by the incident wave packet is shown to decay exponentially [6], i.e., the

scattered wave has the form:

s(t) =∑n

n(t)n +a(t) , n(t) ∼ e−iEnte−�nt

16

If the incident wave packet passes through the scattering region faster than the decay

time τn = 1/�n of a Siegert state, then the outgoing wave can be observed, and it

resembles to a stationary state with the energy En. The more narrow the scattering

resonance is, the longer lives the corresponding Siegert state. So, Siegert states with

�n ≪ 1 may be interpreted as quasi-stationary states of the system that can be excited

by an incident wave and live long after the scattering process is over.

1.3 The Extension to Electromagnetism and its Challenges

One of the goals of this work is to establish in Maxwell’s theory of electromagnetism

a picture similar to that of quantum mechanics in the specific case of TM polarization for

periodic dielectric structures.

Among other differences between the two theories, there is an important distinction

that arises on the physical significance that should be attributed to the Siegert states.

Whereas long lived quantum mechanical Siegert states correspond to the trapping of a

particle in a finite region of space for a very long time, long lived electromagnetic Siegert

states will account for an accumulation of electromagnetic energy in finite regions of

space for a very long time. The subsequent energy build up enhances the intensity

of the electric field in the said regions, thus the enhancement of the nonlinear effects

described by Eq.(1–2).

A major hurdle in the definition of electromagnetic Siegert states is in the analysis of

the eigenvalue problem that leads to the definition of Siegert states in TM polarization.

Following the theory of quantum mechanics, if a solution E(r, t) to the linear wave

equation (1–3) is assumed to have a harmonic time dependence, i.e., E(r, t) =

Eω(r)e−iωt , then one is led to the equation,

H(k2)[Eω] = k2Eω, k2 =

ω2

c2(1–8)

where the operator H(k2) = −�+k2(ε−1) is the analog of the hamiltonian. Note that the

spectral parameter ζ = k2 in this instance does not correspond to the electromagnetic

17

energy, as opposed to the quantum mechanical case where the spectral parameter is

the energy of the particle. The spectral parameter ζ = k2 is simply the square of the

magnitude of a wave vector k in the direction of propagation of the electromagnetic

energy.

An important observation is that the operator H(k2) depends on the spectral

parameter ζ = k2, so that the problem of Eq.(1–8) is a generalized eigenvalue

problem, as opposed to a standard eigenvalue problem such as the one in Eq.(1–5).

If the scattering structure is periodic along a particular direction e (e.g., a periodically

perforated film), the dependence on the spectral parameter ζ = k2 in the generalized

eigenvalue problem (1–8) is further complicated by the boundary conditions imposed

by the scattering of an electromagnetic wave on a periodic structure, i.e., the dielectric

function ε is periodic. The periodicity imposes on the amplitude Eω a condition known as

Bloch’s periodicity condition. It states that if the potential V = k2(ε − 1) has period e,

then,

Eω(r + e) = e ikeEω(r)

for all r and for some real parameter ke . The incorporation of Bloch’s periodicity

condition requires that the generalized eigenvalue problem (1–8) for k2 be studied in

the form,

H′(k2, ke)[Eω] = Eω

where H′(k2, ke) is a compact integral operator with a kernel that is periodic along

the e-axis. While a detailed discussion of this operator at this point is inappropriate

and unnecessary, it is worth mentioning that it is highly nonlinear in k2. In particular,

approaches similar to those in Eqs.(1–7) can not be used to define electromagnetic

Siegert states. Instead, the problem is solved in Chapter 2 using the complex analysis of

compact operators on Hilbert spaces.

A peculiarity that is observed once the Siegert states formalism has been extended

to electromagnetism is the lack of analyticity of solutions to the nonlinear wave equation

18

in the vicinity of spectral, physical and geometrical parameters that allow for the

formation of bound states in the radiation continuum. To be precise, consider the

nonlinear wave equation obtained by keeping the first two terms in Eq.(1–2). It is,

1

c2∂2t(εE + χ2E

2)= �E (1–9)

As already noted, the second order dielectric susceptibility χ2 is only supported on the

scatterers, meaning that it is zero in the vacuum, and it has a certain value χc , assumed

constant, on the scatterers.

Since χc is very small, the textbook approach to solving Eq.(1–9) is to look for a

power series solution in χc as,

E = E1 + χcE2 + χ2cE3 + ...

It turns out that for a periodic system admitting the existence of bound states in the

radiation continuum such a series diverges, indicating a lack of analyticity of the solution

E in χc in the vicinity of a bound state in the radiation continuum. This lack of analyticity

is actually more general. The solution will lack analyticity in other parameters such

as the wave number, and the distance separating two arrays of periodic structures

for instance. A novel fully nonlinear perturbative approach that relies on the theory of

Siegert states is developed to handle this kind of singularity. It is through this theory that

the control of optical nonlinear effects such as second harmonic generation is achieved.

1.4 Main Results

The formalism of Siegert states has been extended to periodic electromagnetic

scattering structures. Results analogous to those of quantum mechanical resonant

scattering theory have been established for electromagnetic resonant scattering theory.

In particular, the Regular Perturbation Theorem for electromagnetic Siegert states has

been formulated and proved. This work appears in Chapter 2 of this Dissertation.

19

By means of the Regular Perturbation Theorem it has been proved that if physical

properties of a scattering periodic structure depend on a physical parameter (e.g. a

distance between two parallel periodic arrays) so that there is a bound state in the

radiation continuum at a particular value of this parameter, then the amplitude of the

field can locally be amplified as much as desired in comparison to the amplitude of the

incident radiation by adjusting the value of the physical parameter. This suggests a novel

universal principle to enhance and control optically nonlinear effects in nanophotonic

periodic structures. The developed methods of electromagnetic Siegert states provide a

novel rigorous mathematical formalism for theoretical studies of such effects. This result

appears in Section 2.2.2 of this Dissertation.

For periodic double arrays of subwavelength dielectric cylinders, it has been shown

that electromagnetic bound states in the radiation continuum exist in the spectral region

where more than one diffraction channels are open. An explicit form of these solutions

of Maxwell’s equations has been found. Qualitatively, the solutions describe wave

guiding modes, either standing or propagating along the structure, which are localized

in a vicinity of the structure and whose wavelength is smaller than the structure period

(the more diffraction channels are open the smaller the wavelength). With increasing

the number of open diffraction channels, wave numbers of these modes become more

sparse. The key difference with commonly known wave guiding modes that occur in

metal cavities and photonic crystals is that the wave numbers of the studied modes lie

in the range of those of radiation (or scattered) modes. These results are presented in

Chapter 3 of this Dissertation.

A novel scattering theory formalism has been developed to study optically

nonlinear effects due to the presence of bound states in the radiation continuum which

resolves the aforementioned problem of non-analyticity of solutions of the nonlinear

electromagnetic scattering problem. This theory is exposed in Section 4.1 of the present

Dissertation.

20

The concept of enhancing optically nonlinear effects in periodic structures

supporting bound states in radiation continuum has been applied to study a resonant

generation of the second harmonics by a periodic double array of parallel dielectric

cylinders with a nonlinear dielectric susceptibility. It has been shown that such a system

is capable of a conversion rate of the fundamental harmonic into the second one as

high as 44%, i.e., comparable to rates achieved by conventional devices. But a striking

contrast to conventional devices, such as, e.g., optically nonlinear crystals, is that: first,

the conversion does not require any phase matching; second, the maximal conversion

rate depends weakly on the nonlinear susceptibility (i.e., the same 44% can be obtained

for a wide range of values of the nonlinear susceptibility); and, third, the device width at

which such a high conversion rate is achieved can be as small as a half of the incident

radiation wavelength. For example, in a typical nonlinear crystal, a similar conversion

rate requires a crystal width of 1-4 cm. For an infrared laser generating radiation with

the wavelength 900 nm, the distance between the two arrays of cylinders at which

the conversion rate of 44% can be achieved is about 450 nm = 4.5×10−5 cm, i.e., the

system width can be reduced by a few thousand times. These properties allow for a

novel principle of miniaturization of nonlinear photonic devices for frequency conversion.

A US patent application is pending for this result (Application Serial No.: 61/488,971).

The details of this result are the object of Chapter 4 of this Dissertation.

21

CHAPTER 2ELECTROMAGNETIC SIEGERT STATES IN PERIODIC STRUCTURES

In this chapter, the theory of Siegert states is extended to Maxwell’s theory of

electromagnetism in TM polarization. In quantum mechanics, the Siegert states are

outgoing solutions to the Schrodinger equation, and they correspond to the positions

of scattering resonances. In the theory of electromagnetism for periodic dielectric

structures, it is found that Siegert states are generalized eigenfunctions to a generalized

eigenvalue problem in which the analog of the Hamiltonian of the system depends on

the eigenvalues of the Siegert states.

2.1 Electromagnetic Siegert States

Consider a scattering problem for an electromagnetic plane wave impinging a

structure made of a non-dispersive dielectric. The structure is described by a dielectric

function ε(r) that defines the value of the dielectric constant at every point r occupied by

the structure and it equals the dielectric constant of the surrounding medium otherwise.

Without loss of generality, the surrounding medium is assumed to be the vacuum, i.e.,

ε = 1. The material is said to be non-dispersive if its dielectric constant does not depend

on the frequency of the incident wave. The structure is assumed to have a translational

symmetry along a particular direction. In this case the dielectric function is independent

of one of the spatial coordinates, say, the y coordinate. If the incident wave is polarized

along the y−axis (the electric field is parallel to the y−axis), then the scattering problem

can be formulated as the scalar Maxwell’s equation:

ε

c2∂2t E = �E , � = ∂2x + ∂2z (2–1)

where c is the speed of light in the surrounding medium (the vacuum), and E is the

electric field. For a planar structure, the function ε differs from 1 only within a strip

(x , z) ∈ (−∞,∞)× (−a, a). A planar structure is periodic if ε(x + Dg, z) = ε(x , z) where

Dg is the period. In what follows, the units of length are chosen so that Dg = 1. It is

22

further assumed that the structure’s dielectric constant exceeds that of the surrounding

medium, i.e., ε(r) ≥ 1 for all r, and ε is bounded. Two examples of such structures

are shown in Fig. 2-1 (Panels (a) and (b)) depicting periodic arrays of infinite dielectric

cylinders, all parallel to the y−axis. The function ε(r) is piecewise continuous. For

general periodic structures, ε is invariant under the action of a suitable affine Coxeter

group in the xy−plane. In this case, vector Maxwell’s equations should be used as the

polarization of the incident wave is not preserved in the scattering process. This general

problem will not be studied here.

If the incident wave has a fixed frequency, then the field E has a harmonic time

dependence E(r, t) = Eω(r)e−iωt in Eq. (2–1) where the amplitude Eω satisfies the

equation:

�Eω(r) + k2ε(r)Eω(r) = 0, k2 =ω2

c2(2–2)

Let the x , y , and z axes be oriented by the unit vectors e1, e2, and e3, respectively.

In the scattering theory, Eω is sought as a superposition of an incident wave, and the

corresponding scattered wave E sω:

Eω(r) = e ik·r + E sω(r), k = kxe1 + kze3 (2–3)

where k is the wave vector of the incident wave. The units of the field E are chosen so

that the amplitude of the incident wave is 1. The scattered wave obeys the outgoing

wave boundary conditions at the spatial infinity. The periodicity of the scattering

structure requires that the amplitude Eω satisfies Bloch’s periodicity condition,

Eω(x + 1, z) = e ikxEω(x , z) (2–4)

Under the specified boundary conditions, the amplitude Eω satisfies the Lippmann-Scwinger

integral equation:

Eω(r) = H[Eω](r) + e ik·r (2–5)

23

where H is the integral operator defined by,

H[E ](r) =k2

4π

∫R2

(ε(r0)− 1)Gk(r|r0)E(r0)dr0

The kernel Gk(r|r0) is the Green’s function for the Helmholtz equation, (� + k2)Gk(r|r0) =

−4πδ(r − r0), with the outgoing boundary condition. It is given by Gk(r|r0) = iπH0(k |r −

r0|) where H0 is the Hankel function of the first kind of order 0. For a proof of the

Lippmann-Schwinger integral equation refer to Section A.1.

Figure 2-1. Panel (a): An example of scattering structures considered in this work.Infinitely long parallel cylinders are placed parallel to a y -axis periodicallyalong an x-axis in the vacuum ε0 = 1. The cylinders are characterized by adielectric constant εc > 1. The rectangle D in Eq.(2–6) is enclosed by thedashed line, and the z-axis.Panel(b): Two periodic arrays such as the one on Panel (a) are placedparallel to each other at a distance 2h between the axes of any two opposingcylinders.Panel (c): The Ckx plane. The cuts run vertically from the diffractionthresholds (indicated by empty circles on the real axis) into the lower half ofthe complex plane. The diffraction thresholds divide the real axis into acountable set of intervals denoted Il , l ≥ 0. The interval I0 lies below theradiation continuum. The rest of the intervals partition the radiationcontinuum.

The existence of solutions to the Lippmann-Schwinger integral equation that also

satisfy Bloch’s condition is established by extending the operator valued function k2 7→

H(k2) to a suitably cut complex plane, and thereafter, applying the Fredholm theory of

24

compact operators. To this end, let the x−component kx of the incident wave vector k be

fixed. The magnitude k = |k| varies only in its z−component, kz = ±√k2 − k2x . Now let

Sε be the support of the function (x , z) → (ε(x , z) − 1) in the strip S = [0, 1] × (−∞,∞)

of the x , z-plane, and let D be a rectangle in S containing Sε, i.e.,

Sε ⊂ D = [0, 1]× [z−, z+] (2–6)

Then for an amplitude Eω satisfying Bloch’s condition,

H[Eω](r) =k2

4π

∑m∈Z

e imkx

∫D

(ε(r0)− 1)Gk(r|r0 −me1)Eω(r0)dr0 (2–7)

where Z denotes the set of all integers. This equation defines H as an operator on

L2(D), the Hilbert space of square integrable functions on D with respect to the

Lebesgue measure. Note that H does not depend on the rectangle D which to some

extent is arbitrary. Indeed, the integrals of Eq.(2–7) extend only over the support Sε of

the function (x , z) 7→ (ε(x , z)− 1) in the strip S = [0, 1]× (−∞,∞) of the x , z-plane. The

rectangle D is only introduced to obtain a connected and compact region of integration,

which is convenient for the subsequent analysis. In general, the support of the function

(x , z) 7→ (ε(x , z)−1) is not necessarily connected as, for example, in the case of multiple

scatterers in the strip S depicted in Fig. 2-1(b). The solution to the Lippmann-Schwinger

integral equation (2–5) will be sought in the Hilbert space L2(D). Such a solution then

extends naturally to the full x , z−plane by Bloch’s periodicity condition (2–4), and the

Lippmann-Schwinger integral equation.

It is not hard to see that the summation and integration can be interchanged in

Eq.(2–7) even though the underlying series is only conditionally convergent. The

Poisson summation formula is then applied to yield the following form for the integral

operator H,

H[E ](r) =

∫D

(ε(r0)− 1)H(k2; r − r0)E(r0)dr0, E ∈ L2(D) (2–8a)

25

where the function H is given by,

H(ζ; x , z) =iζ

2

∑m∈Z

e i(xkx ,m+|z |√

ζ−k2x ,m)√ζ − k2x ,m

, kx ,m = kx + 2πm, ζ = k2x ,m, m ∈ Z (2–8b)

The square roots are defined by choosing the branch cut of the logarithm along the

negative imaginary axis, i.e., if w is a nonzero complex number, then

logw = ln |w |+ i argw , −π2< argw <

3π

2(2–9)

In the scattering theory, the branch points k2x ,m, m ∈ Z, are called the diffraction

thresholds, and they will be referred to as such in what follows.

Note that for w real, then Im{√w} ≥ 0. Therefore, Im

{√ζ − k2x ,m

}> 0 for all but

a finite number of diffraction thresholds k2x ,m such that ζ = k2 > k2x ,m. It follows that in

the series of Eq.(2–8b) all but a finite number of terms decay exponentially as |z | → ∞.

Hence the said series converges uniformly on D. The uniform convergence on D still

holds when the range of the variable ζ = k2 is extended to the cut complex plane Ckx

obtained by excluding all the vertical half lines running from the diffraction thresholds

into the lower half of the complex plane, i.e.,

Ckx = C−∪m∈Z

{k2x ,m − is, s ≤ 0}

This is because in this case too, Im{√

ζ − k2x ,m}> 0 except for a finite number of terms.

It follows that ζ 7→ H(ζ; r) extends to an analytic function on Ckx . Figure 2-1(c) shows a

sketch of the cut plane Ckx .

Since for all ζ ∈ Ckx , the kernel (r, r0) 7→ (ε(r0)−1)H(ζ; r− r0) is square integrable on

D × D, the integral operator H(ζ) is Hilbert-Schmidt in L{L2(D)}, the space of bounded

linear operators on L2(D). In particular, H(ζ) is compact for each ζ ∈ Ckx . Moreover,

the operator valued function defined on Ckx by ζ 7→ H(ζ) is analytic. By the analytic

Fredholm theorem [7], it then follows that if the inverse operator [1 − H(ζ)]−1 exists at

some point ζ ∈ Ckx , it must be meromorphic throughout Ckx . That the said inverse

26

exists at some point is obvious. Indeed, as ζ → 0 in Ckx , the norm of the operator H(ζ),

i.e., ||H(ζ)|| = ||(ε(·) − 1)H(ζ; ·)||L2(D×D), also converges to zero. In particular for ζ

near 0, ||H(ζ)|| < 1 and therefore [1 − H(ζ)]−1 exists as a Neumann series. Thus, the

generalized resolvent ζ 7→ [1− H(ζ)]−1 is meromorphic in Ckx .

The poles {ζn}n of [1 − H(ζ)]−1 are isolated and form a discrete set. The same

analytic Fredholm theorem guarantees that at each of the poles ζn, there exists a

nonzero solution to the generalized eigenvalue problem

H(ζn)[En] = En

The generalized eigenfunctions En will be referred to as the Siegert states.

It will be assumed throughout the rest of the work that the Siegert states are

nondegenerate, i.e., the poles ζn are all simple. The results to be derived in the rest of

the work could easily be generalized to the case when the poles are not simple. Yet, to

our knowledge, there has been no report of resonance poles of multiplicity greater than

one in either the literature of electromagnetism or that of quantum mechanics. However,

we do not have a rigorous proof that higher multiplicity poles cannot occur in the studied

scattering problem. For the structures depicted in Fig. 2-1, perturbation theory (when the

radius of cylinders is much less than the structures period) shows that all the poles are

simple (Chapter 3).

The assumed nondegeneracy of the Siegert states {En}n implies that the residues

{Hn}n of [1− H(ζ)]−1 at the poles {ζn}n are rank one operators on L2(D). In other words,

if ⟨f , g⟩ =∫Df (r0)g(r0)dr0 is the inner product on the Hilbert space L2(D), then

∀n, ∃φn ∈ L2(D) : ∀ψ ∈ L2(D), Hn[ψ] = ⟨φn,ψ⟩En (2–10)

Since the incident wave Ei(k2; r) = e ik·r is analytic in ζ = k2 on Ckx , it follows that the

amplitude Eω = [1 − H(ζ)]−1[Ei(ζ; ·)] extends to a meromorphic function of the variable

27

ζ = k2 on Ckx . Its partial fraction expansion reads

Eω(r) = e ik·r +∑n

an

k2 − k2n + i�nEn(r) + Ea(k

2; r), an = ⟨φn,Ei(k2; ·)⟩

∣∣∣k2=k2n−i�n

(2–11)

where each pole ζn has been decomposed in its real and imaginary parts as ζn =

k2n − i�n owing to the fact that the imaginary parts of the poles are nonpositive as will

be shown shortly. The remainder Ea is analytic in k2 on the cut plane Ckx . It describes

the so called background or potential scattering, while the Siegert states account for the

resonant scattering.

Since the Siegert states are solutions to the homogeneous linear equation

H(ζ)[E ] = E , they should be normalized in some way in order for the functions φn

in Eq.(2–10) and the coefficients an in Eq.(2–11) to be uniquely defined. A natural

normalization condition will be introduced in Section 2.2.1 for Siegert states that arise as

perturbations of bound states in the radiation continuum.

That all the poles of the generalized resolvent ζ 7→ [1 − H(ζ)]−1 have nonpositive

imaginary parts, i.e., �n ≥ 0, ∀n, follows from a series of observations. First, note that a

Siegert state En corresponding to the pole ζn = k2n − i�n satisfies Eq.(2–2) for k2 = ζn,

and therefore

En�En − En�En − 2i�nε|En|2 = 0

By Green’s theorem, it follows that,

2i�n

∫D′

|En(r0)|2ε(r0)dr0 =∮∂D′

(En(r0)∇En(r0)− En(r0)∇En(r0)) · dn0 (2–12)

where D ′ = [0, 1] × [z1, z2] is any rectangle containing the rectangle D, and n0 is the

outward normal. By Bloch’s periodicity condition, the integrals on the line segments

(x , z) ∈ {0, 1} × [z1, z2] of ∂D ′ are canceled out so that the integral on the right side is to

be carried out only over the line segments (x , z) ∈ [0, 1]× {z1, z2} of ∂D ′. The result can

be expressed in terms of the scattering amplitudes S±m associated with the asymptotic

28

behavior of the Siegert state En:

En(r) =

∑m∈Z

S−me

i(xkx ,m−z√

ζn−k2x ,m), z < z−

∑m∈Z

S+me

i(xkx ,m+z√

ζn−k2x ,m), z > z+

(2–13a)

Equations (2–13a) follow immediately from equations (2–8). In particular, the amplitudes

S±m are given by the formulas,

S±m =

iζn

2√ζn − k2x ,m

∫D

(ε(r0)− 1)En(r0)ei(−x0kx ,m∓z0

√ζn−k2x ,m)dr0, r0 = x0e1 + z0e3 (2–13b)

Evaluating the right hand side of Eq.(2–12) yields,

�n =

∑m∈Z Re

{√ζn − k2x ,m

}(|S+

m |2e−2z2Im

{√ζn−k2x ,m

}+ |S−

m |2e2z1Im

{√ζn−k2x ,m

})∫D′ |En(r0)|2ε(r0)dr0

(2–14)

The series in Eq.(2–14) is then split into the sums over the two complementary index

sets,

Iop(ζn) = {m ∈ Z | Re{ζn} ≥ k2x ,m}, Icl(ζn) = {m ∈ Z | Re{ζn} < k2x ,m} (2–15a)

In the scattering theory, when ζ = k2 is real, the sets Iop(k2) and Icl(k2) label open and

closed diffraction channels respectively; thus the superscripts “op” for open, and “cl” for

closed.

The choice of the branch cut for the logarithm given in Eq.(2–9) ensures that

Im{√

ζn − k2x ,m

}> 0, ∀m ∈ Icl(ζn) (2–15b)

In particular, in the series of Eq.(2–14), the contributions from terms whose indices

lie in Icl(ζn) decay exponentially as z2 → +∞ and z1 → −∞. In the said limits, the

aforementioned series is therefore reduced to a finite sum over the set Iop(ζn):

�n = limz1→−∞z2→+∞

∑m∈Iop(ζn) Re

{√ζn − k2x ,m

}(|S+

m |2e−2z2Im

{√ζn−k2x ,m

}+ |S−

m |2e2z1Im

{√ζn−k2x ,m

})∫D′ |En(r0)|2ε(r0)dr0

29

Observing that Re{√

ζn − k2x ,m}≥ 0 for all m ∈ Iop(ζn), and since the dielectric function ε

is positive, it follows that �n ≥ 0, and therefore the imaginary part of the pole ζn is indeed

necessarily negative or zero, i.e., all the poles ζn are in the lower half of the cut complex

plane Ckx .

2.2 Regular Perturbation Theory

Suppose that the dielectric function ε depends on a real parameter h which is

called a coupling parameter. The simplest example of such a coupling is given by two

periodic arrays of dielectric scatterers that are parallel and separated by the distance

2h (Fig. 2-1(b)). The arrays are embedded in a medium of dielectric susceptibility 1.

Each array is characterized by a dielectric function εi(x , z), i = 1, 2 such that εi ≥ 1 on

the scatterers, and εi is of period 1 in the x-direction. The resulting dielectric function

describing the scattering of light on the structure is then,

ε(h; x , z) = 1 + (ε1(x , z − h)− 1) + (ε2(x , z + h)− 1) (2–16)

In the most general case, the dependence of the dielectric function ε on the

coupling parameter h implies that the poles of the generalized resolvent ζ 7→ [1 −

H(h, ζ)]−1 also depend on h. If the coupling is sufficiently smooth as indicated in

the theorem stated below, then the poles of the generalized resolvent as well as the

corresponding Siegert states depend continuously on h.

Before stating the theorem, a clarification must be made on the boundary of the

rectangle D in Eq.(2–6). This rectangle was chosen to contain the support Sε of the

function (x , z) 7→ (ε(x , z) − 1) in the strip S = [0, 1] × (−∞,∞) of the x , z-plane.

In the current situation, the dielectric function depends on the coupling parameter h,

and therefore the support Sε(h) in question could change with h resulting in a different

choice for the set D. This is the case, for instance, in the example of the two parallel

arrays separated by the distance 2h. If the two scatterers in the strip S are taken further

apart, the rectangle D is stretched further accordingly. So, in what follows it will always

30

be assumed that h varies in an open possibly finite interval J0 such that∪

h∈J0 Sε(h) is

bounded, and the rectangle D will be chosen to contain the latter set. The Theorem on

the regular perturbation of electromagnetic Siegert states is formulated as follows:

Regular perturbation theorem. Suppose that the map (h, ζ) 7→ H(h, ζ) from J0 ×Ckx to

L{L2(D)} is continuously Frechet differentiable. Further, suppose that for some hn ∈ J0,

the generalized resolvent ζ 7→ [1 − H(hn, ζ)]−1 has a simple pole ζn ∈ Ckx , and let En be

the corresponding Siegert state. Then there exists an open interval J ⊂ J0 containing hn,

and a unique continuously differentiable function h 7→ ζn(h), h ∈ J, such that ζn(hn) = ζn,

and for all h ∈ J, ζn(h) is a simple pole of the generalized resolvent ζ → [1 − H(h, ζ)]−1.

If En(h) is the Siegert state corresponding to the pole ζn(h), then the function h 7→ En(h)

from J to L2(D) is continuously differentiable.

This theorem is an extension of the Kato-Rellich theorem [5] to the present case.

Due to its length and complexity, the proof is left to the Appendix. It should be noted,

however, that if the functions (x , z) 7→ εi(x , z), i = 1, 2 in Eq.(2–16) are piecewise

differentiable, then (h, ζ) 7→ H(h, ζ) is Frechet differentiable so that the theorem does

indeed hold for two parallel arrays separated by the distance h.

2.2.1 Perturbation of Bound States in the Radiation Continuum

The diffraction thresholds k2x ,m on the real line depend on the x−component kx of

the incident wave vector. They can be ordered independently of the parameter kx by

defining a sequence ζ∗±m(kx) = (2πm ± |[kx ]|)2, m ∈ {0, 1, 2, ...}, where [kx ] is the

argument of e ikx in the interval (−π,π]. The sequence {ζ∗±m(kx)}∞m=0 coincides with the

sequence of diffraction thresholds {(kx + 2πm)2}m∈Z, and for all kx ,

ζ∗0(kx) ≤ ζ∗−1(kx) ≤ ζ∗1(kx) ≤ ζ∗−2(kx) ≤ ζ∗2(kx) ≤ ζ∗−3(kx) ≤ ...

The threshold ζ∗0(kx) is called the radiation continuum threshold, and the interval

I0 = (−∞, ζ∗0(kx)) is said to lie below the radiation continuum. This is because

whenever k2 < ζ∗0(kx), then the scattered amplitude E sω in Eq.(2–3) necessarily decays

31

exponentially in the spatial infinity, |z | → ∞, as can be inferred from Eqs.(2–3), (2–5),

and (2–8). Hence, no electromagnetic flux is carried to the spatial infinity in this spectral

range. In contrast, if k2 > ζ∗0(kx), the amplitude E sω oscillates and represents outgoing

(scattered) radiation that carries an electromagnetic flux to the spatial infinity. The

spectral range above ζ∗0(kx) on a real line is therefore referred to as the radiation contin-

uum. It is the disjoint union of the intervals,

I1 = (ζ∗0(kx), ζ∗−1(kx)), I2 = (ζ∗−1(kx), ζ

∗1(kx)), I3 = (ζ∗1(kx), ζ

∗−2(kx)), ...

Note that when [kx ] is 0 or π, some of the intervals Il are empty, owing to the fact that

some of the diffraction thresholds fuse.

Returning to Siegert states, recall that these states are generalized eigenfunctions

to the generalized eigenvalue problem H(ζn)[En] = En on L2(D) with complex

eigenvalues ζn. These states are naturally extended to the whole x , z−plane by means

of Eqs. (2–13). In general, if the generalized eigenvalue ζn = k2n − i�n lies below or in the

interval I0 in the Ckx -plane, then the corresponding Siegert state is square integrable on

the strip S = [0, 1] × (−∞,∞) of the x , z-plane. This is because for such states, the set

Iop(ζn) of Eq.(2–15a) is empty, and therefore by Eqs.(2–13a) and (2–15b), these states

decay exponentially in the asymptotic region |z | → ∞. In particular, the Siegert states

for which the pole ζb is real and less than the continuum threshold ζ∗0(kx) are the bound

states below the radiation continuum of the system.

On the contrary, the Siegert states En whose corresponding generalized eigenvalues

ζn = k2n − i�n lie below an interval Il , l ≥ 1 of the radiation continuum, i.e., k2n ∈ Il and

�n > 0, are not necessarily square integrable on the strip S . This is because for

these states, the set Iop(ζn) is not empty, and consequently, the terms of the series in

Eq. (2–13a) indexed by this set are unbounded. However, when the pole ζn is in Il , l ≥ 1,

i.e., �n = 0, it will be shown shortly that the resulting Siegert state is square integrable

32

on the strip S and therefore, such a Siegert state is a bound state in the radiation

continuum. The rest of this subsection is devoted to the proof of this assertion.

To proceed, suppose that for some value hb of the coupling constant h, there exists

a Siegert state Eb whose corresponding generalized eigenvalue ζb is real and lies in an

interval Il , l ≥ 1, of the radiation continuum. By the Regular Perturbation Theorem there

exist continuously differentiable functions h 7→ ζn(h) and h 7→ En(h) on an interval J

containing hb such that ζn(hb) = ζb, and En(hb) = Eb. Furthermore, En(h) is the Siegert

state corresponding to the pole ζn(h) of the generalized resolvent ζ 7→ [1 − H(h, ζ)]−1

for all h ∈ J. Put ζn(h) = k2n (h) − i�n(h) as before. Then �n(hb) = 0. Without loss of

generality, the interval J is assumed to be sufficiently small so that for all h ∈ J\{hb},

�n(h) = 0. Equation (2–14) for the Siegert state En(h) is then rewritten as∫D′

|En(r0)|2ε(h; r0)dr0 =1

�n

∑m∈Z

Re{√

ζn − k2x ,m

}(|S+

m |2e−2z2Im

{√ζn−k2x ,m

}

+ |S−m |2e

2z1Im{√

ζn−k2x ,m

})

where it is understood that the Siegert state En, the pole ζn, and its imaginary part −i�n,

as well as the amplitudes S±m are all functions of h. Note that the values of z1 and z2 are

assumed to be independent of the coupling parameter h. This is because h varies in a

small interval J, and therefore the values of z± in Eq.(2–6), which now depend on h are

bounded. The condition z1 ≤ z−(h) < z+(h) ≤ z2, ∀h ∈ J, can therefore be realized by

choosing z1 and z2 sufficiently large.

As h → hb, then En → Eb in L2(D), and therefore the left hand side of Eq.(2–11)

remains finite as the dielectric function ε(h; ·) is bounded. The limit of the right hand

side as h → hb may also be calculated by first computing explicitly the complex square

roots involved according to the logarithmic branch cut described in Eq.(2–9). Put

33

ξm(h) = k2n (h)− k2x ,m, m ∈ Z. Then

√ζn(h)− k2x ,m =

√√ξm(h)2+�n(h)2+ξm(h)

2− i�n(h)√

2(√

ξm(h)2+�n(h)2+ξm(h)) , if m ∈ Iop(ζn(h))

− �n(h)√2(√

ξm(h)2+�n(h)2−ξm(h)) + i

√√ξm(h)2+�n(h)2−ξm(h)

2, if m ∈ Icl(ζn(h))

where Iop and Icl are the index sets of Eq.(2–15a). In particular, if m ∈ Iop(ζn(h)), then

ξm(h) ≥ 0, whereas ξm(h) < 0 whenever m ∈ Icl(ζn(h)). As h → hb, �n(h) → 0, and,

∫D′

|Eb(r0)|2ε(hb; r0)dr0 = limh→hb

1

�n(h)

∑m∈Iop(ζb)

√ξm(h)

(|S+

m(h)|2 + |S−m |2)

−∑

m∈Icl (ζb)

1

2√−ξm(hb)

(|S+

m(hb)|2e−2z2√

−ξm(hb)

+ |S−m(hb)|2e2z1

√−ξm(hb)

)(2–17)

In particular, for each m ∈ Iop(ζb), the limits limh→hb|S±

m (h)|2�n(h)

must exist and must be finite.

By continuity of the functions h 7→ S±m(h) on J, there exist complex numbers S±

m,b such

that

limh→hb

S±m(h)√�n(h)

= S±m,b, ∀m ∈ Iop(ζb) (2–18)

As z1 → −∞ and z2 → ∞, the second series in Eq.(2–17) converges to zero.

Therefore the function r → Eb(r)√ε(hb; r) is square integrable on the strip S =

[0, 1]× (−∞,∞) of the x , z−plane, and∫S

|Eb(r0)|2ε(hb; r0)dr0 =∑

m∈Iop(ζb)

√ξm(hb)

(|S+

m,b|2 + |S−

m,b|2)

(2–19a)

Since, ε(hb, ·) ≥ 1, it follows that Eb ∈ L2(S) as claimed. Thus, if Eb is a bound state in

the radiation continuum, it can always be normalized by the condition∫S

|Eb(r0)|2ε(hb; r0)dr0 = 1 (2–19b)

34

This result justifies the term ”bound state” introduced by analogy with quantum

mechanics where bound states represent square integrable eigenfunctions of the

Hamiltonian operator, i.e., an electromagnetic bound state is a localized solution of

Maxwell’s equations.

Finally, by continuity En(h) → Eb as h → hb, and therefore the normalization

condition (2–19b) determines uniquely the Siegert states En(h) along the curve h 7→

ζn(h), h ∈ J.

2.2.2 Near Field Amplification Mechanism

Resonant phenomena in the scattered electromagnetic flux can be described by

the formalism of Siegert states as given in Eq.(2–11). However, a complete description

requires calculating the residues an in Eq.(2–11). Here these residues are calculated for

Siegert states that arise as perturbations of bound states in the radiation continuum in

the sense of Section 2.2.1, i.e., for the states En(h) where |h − hb|/hb ≪ 1. In particular,

it is shown that if the scattering structure has bound states in the radiation continuum,

then there exist regions (the so called ”hot spots”) in which the field amplitude can be

amplified as much as desired by taking the value of the coupling parameter h close

enough to hb. The unbounded local growth of the field amplitude is essentially due

to the linearity of Maxwell’s equations. It disappears when a non-linear dielectric

susceptibility, required for large field amplitudes, is included into Maxwell’s equations

(Chapter 4). Nevertheless, such a local field amplification enhances quite substantially

optical nonlinear effects in the scattering structure as shown in Chapter 4. The following

analysis also suggests that the concept to enhance optically non-linear effects by using

bound states in the radiation continuum is universal because the existence of ”hot spots”

is proved to be a characteristic feature of such scattering structures.

When estimating the coefficients an, it is convenient first to give another equation for

them that is alternative to that of Eq.(2–11). Next, this equation will be analyzed in the

vicinity of a bound state in the radiation continuum. In particular, if a bound state in the

35

radiation continuum Eb exists at the critical value hb of the coupling parameter, and J is

the interval produced by the Regular Perturbation Theorem, while En(h; r) is the Siegert

state that arises as a continuous perturbation of the bound state Eb for h ∈ J, then it will

be proved that an(h) ∼√�n(h) as h → hb. Recall that ζn(h) = k2n (h) − i�n(h) is the

generalized eigenvalue at which the Siegert state En(h) exists.

To proceed, put Eω = ζ−ζnan

Eω where Eω is the amplitude in Eq.(2–3). Then by the

decomposition (2–11), Eω → En as ζ → ζn. From the system,�Eω + ζε(h)Eω = 0

�En + ζε(h)En = 0

it is derived by Green’s theorem that,∫∂D′

(En∇Eω − Eω∇En

)· dn0 + (ζ − ζn)

∫D′EωEnε(h)dr0 = 0

where D ′ is the same rectangle as in Eq. (2–12). Then, by splitting the amplitude Eω in

terms of the incident and scattered waves, Eω(r) =ζ−ζnan

e ik·r + E sω(r), one infers that

an(h) = −(ζ − ζn)

[∫∂D′(ikEn −∇En)e

ik·r0 · dn+ (ζ − ζn)∫D′ Ene

ik·r0ε(h)dr0]∫

∂D′

(En∇E s

ω − E sω∇En

)· dn+ (ζ − ζn)

∫D′ E s

ωEnε(h)dr0

where it is understood that En and ζn are functions of h. Since an is independent of ζ, the

desired expression for an is obtained by taking the limit ζ → ζn:

an(h) = −∫∂D′

(iknEn −∇En

)e ikn·r0 · dn∫

∂D′

(En∇∂ζE s

ω − ∂ζE sω∇En

)· dn

∣∣∣ζ=ζn

+∫D′ |En|2ε(h)dr0

, kn = k

∣∣∣ζ=ζn

(2–20)

where for the first term in the denominator, l’Hopital’s rule has been applied. This

formula would not generally be useful as the term E sω involves the coefficients an

implicitly. However, if the Siegert state En(h) is taken near a bound state in the radiation

continuum, as assumed here, the first-order perturbative expression of an only depends

on the Siegert state En(h).

36

Such a perturbative expression is obtained by analyzing each of the integrals in

Eq. (2–20) separately. First, it is observed that as h → hb, then En(h) → Eb. By letting

z1 → −∞, and z2 → ∞, it follows from the normalization of Eq.(2–19b) that the second

integral in the denominator of Eq.(2–20) can be made arbitrarily close to 1 provided h is

sufficiently close to hb. Hence, if the first integral of the said denominator can be shown

to converge to zero in the limits considered, this would imply that in the leading order of

perturbation theory, the denominator of Eq. (2–20) is approximated by 1. That the first

integral of the said denominator does indeed converge to zero as h → hb, z1 → −∞,

and z2 → ∞ can be established through a lengthy, but straightforward calculation whose

details are omitted. It can be carried out along the following lines. As noted earlier,

Bloch’s condition implies that the first integral in the denominator of Eq. (2–20) is to be

carried out on the segments (x , z) ∈ [0, 1] × {z1, z2} of the boundary of D ′. Now, as

h → hb, the Siegert state En(h) becomes a bound state Eb, and therefore, it decays

exponentially in the spatial infinity |z | → ∞. Similarly, the same exponential decay can

be established for the terms involving the derivatives of E sω in the limit h → hb. It then

follows that in the limits h → hb, z1 → −∞ and z2 → ∞, the integral in question vanishes.

Thus the denominator of Eq.(2–20) remains indeed close to 1, provided h is sufficiently

close to hb.

On the other hand, the numerator to Eq.(2–20) can be evaluated in terms of the

amplitudes S±m in Eqs.(2–13) to yield∫

D′

(iknEn −∇En

)e ikn·r0 · dn =

√�n(h)An(h, z1, z2)

where if kn = kxe1 +√ζn − k2x e3, then

An(h, z1, z2) = 2iRe{√

ζn(h)− k2x

} S+0 (h)√�n(h)

e−2z2Im

{√ζn(h)−k2x

}

+ 2Im{√

ζn(h)− k2x

} S−0 (h)√�n(h)

e2iz1Re

{√ζn(h)−k2x

}

37

The case kn = kxe1 −√ζn − k2x e3 is similar. As h → hb, then ζn(h) → ζb, and since

ζb > k2x , it follows that Im{√

ζn(h)− k2x

}→ 0. Hence, in the said limit, An(h, z1, z2) →

2iS+0,b

√ζb − k2x for S+

0,b given in Eq.(2–18). Thus, in general, an can be written as

an(h) = an(h)√�n(h) (2–21)

where an(h) is bounded and has the property an(h) → 2iS+0,b

√ζb − k2x as h → hb.

The most important consequence of the structure of the coefficients an near a

bound state in the radiation continuum is a local amplification of the amplitude Eω as

compared to the amplitude of the incident radiation. Indeed, if the wavenumber k of

the incident radiation and the coupling parameter h are tuned to satisfy the condition

k = kn(h), ∀h ∈ J, then the amplitude (2–11) becomes:

Eω(r) = ian(h)√�n(h)

En(r) + Ea(h; r) (2–22)

where h 7→ ||Ea(h, ·)||L2(D) is bounded on J. It follows that,

||Eω||L2(D) ≥

∣∣∣∣∣ |an(h)|√�n(h)

||En||L2(D) − ||Ea||L2(D)

∣∣∣∣∣As h → hb, then En → Eb in L2(D), and therefore ||En||L2(D) → ||Eb||L2(D) = 0. Hence if the

constant S+0,b in Eq.(2–21) is nonzero, then,

limh→hb

||Eω||L2(D) = ∞

and this can only happen if the amplitude Eω diverges in some regions of the rectangle

D. Even though the fact that an(h) is always nonzero in the limit h → hb (i.e., S+0,b =

0) could not be verified for all kinds of couplings, there are many systems in which

it holds true. For example, this is the case of the normal incidence (kx = 0) for a

symmetric double array depicted in Fig. 2-1(b) when the dielectric functions ε1 and ε2 in

Eq.(2–16) are identical, and symmetric with respect to the reflection (x , z) 7→ (x ,−z).

In this particular case, the parity operator, S[E ](x , z) = E(x ,−z) on L2(D), and the

38

Lippmann-Schwinger integral operator H commute so that Siegert states are always

symmetric or skew symmetric with respect to the reflection (x , z) 7→ (x ,−z) in the

x , z-plane. In particular, the amplitudes S±m given in Eqs.(2–13) are such that S+

m = ±S−m

depending on whether the Siegert state they correspond to is symmetric or skew

symmetric. It follows that if the bound state Eb happens to lie in the interval I1 of the

Ckx -plane (case of one open diffraction channel), then Eqs.(2–19) are reduced to

|S+0,b|

2 = |S−0,b|

2 =1

2√ζb − k2x

so that the coefficient in question is indeed nonzero and the amplitude Eω is amplified in

the vicinity of the bound state in the radiation continuum Eb. Note that the amplification

of the amplitude Eω was also observed perturbatively in the case of a more general

incidence angle (i.e., kx = 0) when the bound state Eb lies in the intervals I1 and I2 of the

radiation continuum (Chapter 3).

The aforementioned amplification can only happen in a region near or within the

scattering structure. This can be understood by analyzing the first term of Eq.(2–22)

from which the amplification should result. Outside the scattering region, the Siegert

state is expressed in terms of its scattered amplitudes by Eq.(2–13a). As before, the

series involved in this expressions split over the index sets Iop(ζb) and Icl(ζb) ( for h

near hb, Iop(ζn(h)) = Iop(ζb) and Icl(ζn(h)) = Icl(ζb)). On one hand, the terms of the

series whose indices lie in Icl(ζb) decay exponentially in the spatial infinity so that no

amplification of the field can be obtained from them. On the other hand, the terms

whose indices lie in Iop(ζb), disappear near the bound state in the radiation continuum

as these terms are proportional to√�n(h) as indicated by Eq.(2–18). From the physical

point of view, the noted local field amplification results from a constructive interference

of scattered fields from each scatterer (an elementary cell) of the periodic structure. It

is important to note that the amplification magnitude can be regulated by varying the

coupling parameter h, which provides a natural physical mechanism to control optical

39

nonlinear phenomena if the structure has a nonlinear dielectric susceptibility. This

mechanism has been used to demonstrate that a double periodic array of dielectric

cylinders (depicted in Fig. 2-1 (b)) with a non-zero second-order nonlinear susceptibility

can convert as much as 44% of the incident flux into the second harmonics when the

distance between the arrays is properly adjusted. The smallest distance at which such

a high conversion rate can be achieved is about a half of the wave length of the incident

radiation (Chapter 4).

2.3 Decay of Electromagnetic Siegert States

A Siegert state can be excited by an incident radiation, e.g., by a wave packet

whose spectral distribution peaks around a desired wave number kc ≈ kn. When passing

through the structure, some of the wave packet energy is trapped by the structure and

remains in there for a long period of time, decaying slowly by emitting a monochromatic

radiation. This is established in a fashion similar to that of the study of the decay of

unstable states in quantum mechanics [6]. To illustrate the principle, only the case of

normal incidence is considered here (i.e., kx = 0). Other incidence angles can be

treated in a similar manner.

The exact statement which will be proved is as follows. Suppose that a Siegert state

En exists at the pole ζn = k2n − i�n such that kn > 0 and �n > 0 (i.e., the Siegert state

is not a bound state in the radiation continuum). Then this state will decay with time by

emitting a monochromatic radiation at a wavenumber kn, and with a life time τn:

kn =

√k2n +

√k4n + �2n2

−−−→�n≪1

kn, τn =2knc�n

−−−→�n≪1

2knc�n

(2–23)

As a starting point, consider an incident wave packet,

Ei(r, t) =

∫ ∞

0

A(k) cos(k · r − ωt)dk , ω = ck

40

where A(k) is the distribution of wavenumbers in the wave packet. Then the solution E

of the wave equation (2–1) reads

E(r, t) = Re{∫ ∞

0

A(k)Eω(r)e−iωtdk

}where Eω is the amplitude of Eq. (2–2). From the meromorphic expansion of Eω in

Eq. (2–11) it then follows that,

E(r, t) = Re

{Ei(r, t) +

∑n

En(r)n(t) +

∫ ∞

0

A(k)Ea(k2, r)e−iωtdk

}

where the time dependence n(t) of the Siegert state En is,

n(t) = an√�n

∫ ∞

0

A(k)

k2 − k2n + i�ne−icktdk (2–24)

for an defined in Eq.(2–21). As t → ∞, then n(t) → 0 as required by the Riemann-Lebesgue

lemma. For a typical physical wave packet, the function A(k) decays fast as k → ∞ and

is also analytic in the complex k−plane. These properties allows for evaluating the

integral (2–24) by the standard means of the complex analysis. It is also worth noting

that bound states in the radiation continuum cannot be excited by an incident radiation

because �n = 0.

To avoid excessive technicalities of the general case, a specific form of the

distribution A(k) is chosen to illustrate the procedure, which is sufficient to establish

the main properties of the decay of Siegert states. The simplest form that is also most

commonly used in physics is a Gaussian wave packet centered around a wavenumber

kc :

A(k) =e− (k−kc )

2

2σ2

σ√2π

In the limit σ → 0, A(k) → δ(k − kc), and the monochromatic case is recovered. The

analysis will be carried out for a Siegert state En corresponding to a pole ζn = k2n − i�n

such that k2n > 0 and �n > 0. Siegert states with k2n < 0 are discussed at the end of this

section.

41

The integrand in (2–24) is highly oscillatory for large t, the contour of integration

must be deformed to a curve along which the phase is constant to obtain a fast

converging integral, and thereby to determine the leading term. To this end, first, the

change of variable ξ = k−kc+ictσ2

σ√2

is made to obtain the following more amenable form for

n:

n(t) =an√�ne

−ictkc− 12c2t2σ2

2σ2√π

∫C0

e−ξ2

(ξ − ξ+)(ξ − ξ−)dξ, ξ± =

−kc + ictσ2 ±√ζn

σ√2

(2–25)

where the contour of integration C0 is a horizontal ray outgoing from the point ξ0 =

−kc+ictσ2

σ√2

toward the infinity in the ξ-plane as shown in Fig. 2-2. The position of the poles

ξ± indicated on the same figure follows from the fact that for k2n > 0, then Re{√ζn} > 0

while Im{√ζn} < 0.

Put (u, v) = (Re ξ, Im ξ). Since the function ξ 7→ e−ξ2 decays exponentially in the

region Re{ξ2} > 0, the contour of integration in Eq.(2–25) can be deformed to a contour

that consists of the constant phase curve C1 = {(u, v) | uv = −kcct2, u ≤ Re ξ0} extending

from ξ0 to −∞ and another constant phase contour C2 : Im{ξ} = 0 from −∞ to ∞.

Figure 2-2 shows the modified contour. By Cauchy’s theorem, n becomes the sum of

three terms:

n(t) = an(An(t) + Bn(t) + Cn(t))

where An is the residual contribution at the pole ξ+, while Bn and Cn are the contributions

of the line integrals along the contours C1 and C2, respectively. It is proved shortly that

it is the residual term An that accounts for an exponential time decay of the Siegert

state En. For a long period of time, this term dominates in the decay radiation of a

Siegert state, and it is only after it has decayed considerably that the term Bn becomes

dominant. The term Cn remains small in comparison to An and Bn.

To begin, the residual term An is evaluated to yield the formula,

An(t) = πi

√�n√ζnA(√ζn)e

−ict√ζn

42

In particular, the wavenumber kn, and the life time τn of the Siegert state given in

Eq.(2–23) follow from the formula for√ζn which is,

√k2n − i�n =

√k2n +

√k4n + �2n2

− i�n√

2(k2n +

√k4n + �2n

)Note that for �n ≪ 1 (a narrow resonance), the amplitude of the Siegert state is

proportional to A(√ζn) ≈ A(kn). Hence, the Siegert state corresponding to the scattering

resonance at k = kn is excited if the Gaussian wave packet has a sufficient amplitude at

k = kn, or, ideally, is centered at kn to achieve the maximal effect.

The terms Bn and Cn also decay necessarily in time. For Cn, the integral along C2

will necessarily remain bounded in time so that the multiplicative factor in Eq.(2–25) is

dominant, i.e.,

Cn(t) = O(e−12c2t2σ2

)

The same relation holds for Bn, provided that

ct ≤ kc

σ2(2–26)

When this condition is violated, the point ξ0 moves into the region Re{ξ2} < 0, and

the exponential e−ξ2 starts to grow large on a part of the contour C1. Nonetheless,

the term Bn(t) still decays to zero in time. This is because, as noted earlier, the

Riemann-Lebesgue lemma ensures the decay of n(t) in time, and the terms An and Cn

have been proved to converge to zero. For the actual decay rate of Bn, it can be inferred

from Laplace’s method for the asymptotic expansion of integrals that Bn = O(1/tα) for

some positive number α, which is of no importance here however. Indeed, it is obvious

that as t increases, the term Bn should be larger than the residue at ξ+, since the arc of

the curve C1 in the region Re{ξ2} < 0 grows longer, and therefore the exponential decay

of the Siegert state can no longer be observed above the background described by Bn.

The condition (2–26) shows that the smaller the width σ of the wave packet is, the longer

43

the exponential decay can be observed. For instance, one can ensure the observation

of the decay radiation of the state En by requiring that the half-life time occurs before the

decay amplitude becomes smaller than that of the background, i.e.,

cτn ≤kc

σ2⇐⇒

ln 2√k2n +

√k4n + �2n

kc≤ �nσ2

In other words, the width σ of the packet must be comparable to√�n, if not smaller.

Figure 2-2. The ξ-plane for the integral in Eq. (2–25). The region Re{ξ2} ≥ 0 isshadowed. The exponential e−ξ2 is bounded in it. The sketch is realizedunder the condition (2–26), when the point ξ0 is still in the shadowed region.When the condition is violated, part of the curve C1 lies in the regionRe{ξ2} < 0.

For Siegert states with k2n < 0, the pole ξ+ is no longer enclosed by the contours

C0, C1, and C2 if t is sufficiently large. The pole moves to the left of the contour C1 and

the exponential time decay is absent, owing to the fact that the incident wave packet

does not contain any radiation at wavenumbers close to kn for the Siegert state to be

excited.

2.4 Proof of the Regular Perturbation Theorem

Recall that ∀(h, ζ) ∈ J0 ×Ckx , H(h, ζ) ∈ L{L2(D)}, and that the map

J0 ×Ckx → L{L2(D)}

(h, ζ) 7→ H(h, ζ)

44

is continuously Frechet differentiable. For (h, ζ) ∈ J0 × Ckx , let Rλ(H(h, ζ)) = [λ −

H(h, ζ)]−1, λ ∈ C, be the resolvent of H(h, ζ) when it exists, and let ρ(H(h, ζ)) =

{λ ∈ C|Rλ(H(h, ζ)) exists} be the resolvent set of H(h, ζ). The proof of the Regular

Perturbation Theorem is aided with the following lemma:

Lemma 1. Let S = {(h, ζ,λ) ∈ J0 × Ckx × C|λ ∈ ρ(H(h, ζ))}. The set S is open in

J0 ×Ckx ×C and nonempty.

Proof. That S is nonempty follows from the results of Section 2.1. Indeed, if ζ is not a

pole of ζ 7→ [1 − H(h, ζ)]−1, then (h, ζ, 1) ∈ S. To show that S is open, let (h0, ζ0,λ0) ∈

S. Then one has to prove that there always exists a neighborhood of (h0, ζ0,λ0) in

J0 ×Ckx ×C that lies in S.

Since ρ(H(h0, ζ0)) is open in C, there exists λ1 ∈ ρ(H(h0, ζ0)), λ1 = λ0 so that

[λ1 − H(h0, ζ0)]−1 exists. Let then be the function,

: J0 ×Ckx → L{L2(D)}

(h, ζ) 7→ λ1 − H(h, ζ)

The map is continuous, and (h0, ζ0) is invertible. As the set of all invertible operators

in L{L2(D)} is open, there exists an open neighborhood U ⊂ J0 × Ckx of (h0, ζ0) such

that ∀(h, ζ) ∈ U, (h, ζ) is invertible.

Now by the first resolvent formula,

Rλ0(H(h0, ζ0))− Rλ1(H(h0, ζ0)) = (λ1 − λ0)Rλ0(H(h0, ζ0))Rλ1(H(h0, ζ0))

so that,

Rλ0(H(h0, ζ0))(1− (λ1 − λ0)Rλ1(H(h0, ζ0))

)= Rλ1(H(h0, ζ0))

It follows that 1− (λ1 − λ0)Rλ1(H(h0, ζ0)) is invertible. Now let � be the function,

� : U ×C→ L{L2(D)}

((h, ζ),λ) 7→ 1− (λ1 − λ)Rλ1(H(h, ζ))

45

Then � is continuous. Since �(h0, ζ0,λ0) is invertible, and again, the set of all invertible

operators in L{L2(D)} is open, it follows that there exists an open set W ⊂ U ×C such

that for all (h, ζ,λ) ∈ W , the operator �(h, ζ,λ) is invertible.

Thus, for (h, ζ,λ) ∈ W , both λ1 − H(h, ζ) and 1− (λ1 − λ)Rλ1(H(h, ζ)) are invertible.

Their composition is therefore invertible, and this composition is,

(λ1 − H(h, ζ))(1− (λ1 − λ)Rλ1(H(h, ζ))) = λ− H(h, ζ)

It follows that W ⊂ S. As W is a neighborhood of (h0, ζ0,λ0) in J0 ×Ckx ×C, the set S is

open.

The proof of the Regular Perturbation Theorem is as follows.

Proof. Let ζn be a simple pole of ζ 7→ [1−H(hn, ζ)]−1 for some fixed hn ∈ J0, and let En be

the corresponding Siegert state. Then 1 is an eigenvalue of H(hn, ζn) corresponding to

the eigenfunction En in the usual sense. The theorem amounts to proving the existence

of a curve h 7→ ζ(h) in J0 × Ckx along which the family of operators H(h, ζ(h)) still has

1 as an eigenvalue. The corresponding eigenfunctions will then be the Siegert states of

the said operators along the curve in question.

The proof starts by providing a general formula for the eigenvalue λ0(h, ζ) of H(h, ζ)

which is the perturbed value of the eigenvalue 1 when the point (hn, ζn) is displaced to

(h, ζ) in the J0 ×Ckx space. Then by the Implicit Function Theorem, a curve h 7→ ζ(h) is

found along which the eigenvalue λ(h, ζ) remains 1, i.e., λ(h, ζ(h)) = 1.

As a starting point, note that as H(hn, ζn) is a compact operator on the Hilbert

space L2(D), the Riesz-Schauder theorem [7] implies that 1 is necessarily an isolated

eigenvalue. Therefore, there exists δ > 0 such that 1 is the only eigenvalue of H(hn, ζn)

in the disk {λ ∈ C | |λ− 1| ≤ δ}. It follows that Vδ = {(hn, ζn,λ)|λ ∈ C, |λ− 1| = δ} ⊂ S}

where S is the set of the previous lemma. Since S is open, and Vδ is compact, there

exists an open W in J0 × Ckx × C such that Vδ ⊂ W ⊂ S. Therefore there exists

46

a connected neighborhood U of (hn, ζn) in J0 × Ckx such that λ ∈ ρ(H(h, ζ)) for all

(h, ζ) ∈ U, and for all λ ∈ C with |λ− 1| = δ.

Put

P(h, ζ) =1

2πi

∮|λ−1|=δ

[λ− H(h, ζ)]−1dλ, (h, ζ) ∈ U (2–27)

Then P(h, ζ) ∈ L{L2(D)}, and by Theorem XII.6 of [5], P(h, ζ) is a projection. As the

map (h, ζ) 7→ P(h, ζ) of U to L{L2(D)} is continuous, it follows that the dimension of the

range of P(h, ζ) is constant throughout U.

Since the eigenvalue 1 of H(hn, ζn) is nondegenerate, it follows that the range of

P(hn, ζn) has dimension 1, and therefore the dimension of the range of P(h, ζ)) is 1

throughout U. By Theorem XII.6 of [5], it follows that for all (h, ζ) ∈ U, there exists a

unique eigenvalue λ0(h, ζ) of H(h, ζ) in {λ ∈ C | |λ− 1| ≤ δ}, and P(h, ζ) is the projection

on the corresponding eigenspace. In particular, P(h, ζ)[En] is in the eigenspace of

λ0(h, ζ), and therefore,

H(h, ζ)[P(h, ζ)[En]

]= λ0(h, ζ)P(h, ζ)[En], (h, ζ) ∈ U

Hence,

λ0(h, ζ) =⟨En, H(h, ζ)

[P(h, ζ)[En]

]⟩

⟨En, P(h, ζ)[En]⟩, (h, ζ) ∈ U

where once again, ⟨·, ·⟩ is the inner product on L2(D). This is the formula for the

perturbed eigenvalue announced in the preamble of this proof.

Now, observe that since (h, ζ) 7→ H(h, ζ) is continuously Frechet differentiable

in U, so is (h, ζ) 7→ P(h, ζ). Therefore, λ0 is continuously Frechet differentiable in U.

Now, λ0(hn, ζn) = 1, and as it will be shown shortly, ∂ζλ0(hn, ζn) = 0. It follows by the

Implicit Function Theorem for Banach spaces that there exists an open interval J ⊂ J0

containing hn, and a unique continuously differentiable function h 7→ ζn(h), h ∈ J,

such that ζn(hn) = ζn and λ0(h, ζn(h)) = 1 for all h ∈ J. Thus the proof of the Regular

perturbation theorem is complete, provided the property ∂ζλ0(hn, ζn) = 0 is established.

47

The latter is achieved by studying the analyticity of the resolvent Rλ(H(hn, ζ)) = [λ −

H(hn, ζ)]−1 in the variables λ and ζ separately when h = hn is fixed.

Let λ ∈ C and ζ ∈ Ckx such that |λ − 1| = δ, (hn, ζ) ∈ U, and ζ = ζn. Then the

resolvent Rλ(H(hn, ζ)) = [λ− H(hn, ζ)]−1 exists by definition of the neighborhood U. Also,

the resolvent R1(H(hn, ζ)) = [1 − H(hn, ζ)]−1 exists and is meromorphic in ζ. By the first

resolvent formula,

Rλ(H(hn, ζ))− R1(H(hn, ζn)) = (1− λ)Rλ(H(hn, ζ))R1(H(hn, ζ)) (2–28)

Substituting Eq.(2–28) into Eq.(2–27), it follows that,

P(hn, ζ) =1

2πi

∮|λ−1|=δ

(1− λ)Rλ(H(hn, ζ))R1(H(hn, ζ))dλ (2–29)

Now, the meromorphic expansion of λ 7→ Rλ(H(hn, ζ)) at λ0(hn, ζ) is,

Rλ(H(hn, ζ)) =1

λ− λ0(hn, ζ)P(hn, ζ) + Rλ(hn, ζ) (2–30)

where λ 7→ Rλ(hn, ζ) is analytic in the disk {λ ∈ C : |λ − 1| ≤ δ}. The substitution of

Eq.(2–30) into Eq.(2–29) yields

P(hn, ζ) = (1− λ0(hn, ζ))P(hn, ζ)R1(H(hn, ζ)) (2–31)

Now, at the pole ζn, the generalized resolvent ζ 7→ R1(hn, ζ) has the meromorphic

expansion:

R1(hn, ζ) =1

ζ − ζnHn + R(hn, ζ) (2–32)

where Hn is the residue of ζ 7→ R1(hn, ζ) as given in Eq.(2–10), and ζ 7→ R(hn, ζ) is

analytic in the vicinity of ζn. Substituting Eq.(2–32) into Eq.(2–31) and taking the limit as

ζ → ζn yields the equation,

P(hn, ζn) = −∂ζλ0(hn, ζn)P(hn, ζn)Hn (2–33)

48

Next, the action of both the sides of this operator equality is evaluated on the state En.

Since P(hn, ζn) projects on En, and in light of Eq.(2–10), it follows that,

En = −⟨φn,En⟩∂ζλ0(hn, ζn)En

As the Siegert state En is not identically zero, it follows that ∂ζλ0(hn, ζn) = 0, and the

proof of the theorem is complete.

A final noteworthy remark is that the projection P(hn, ζn) and the residue Hn are

proportional, which is established by applying both the sides of Eq.(2–33) to an arbitrary

state F ∈ L2(D):

P(hn, ζn)[F ] = −∂ζλ0(hn, ζn)Hn[F ]

49

CHAPTER 3ELECTROMAGNETIC BOUND STATES IN THE RADIATION CONTINUUM FOR

PERIODIC DOUBLE ARRAYS OF SUBWAVELENGTH DIELECTRIC CYLINDERS

In this chapter, bound states in the radiation continuum are studied analytically for

a system of two arrays of parallel dielectric cylinders. The approach is based on the

resonant scattering theory [5, 10] where the bound states are identified as resonances

with the vanishing width (the distance between the arrays is a physical parameter

which regulates the coupling of the resonances). The study is carried out for the whole

radiation spectrum range. In addition to the bound states in the zero-order diffraction

(which were known to exist from the early numerical studies), bounds states are found in

the spectral range where two or more diffraction channels are open. Analytic solutions

of Maxwell’s equations for all the bound states are given in the limit when the cylinder

radius is much smaller than the period of the structure. The system is shown to have

bound states below the radiation continuum whose explicit form is also found.

3.1 Scattering Theory and Classification of the Fields

The system considered is sketched in Fig. 3-1(a). It consists of an infinite double

array of parallel, periodically positioned, dielectric cylinders suspended in the vacuum

[11]. The cylinders are assumed to be non-dispersive with a dielectric constant εc > 1.

The coordinate system is set so that the cylinders are parallel to the y−axis, the

structure is periodic along the x−axis, and the z−axis is normal to the structure.

The unit of length is taken to be the array period and the arrays have a relative mismatch

a ∈ [0, 12 ] along the x-axis.

The structure is illuminated by a linearly polarized monochromatic beam with the

electric field parallel to the cylinders (TM polarization). In the given settings, Maxwell’s

equations are reduced to the scalar wave equation for a single component of the electric

field, denoted E , in the x , z−plane:

ε

c2∂2t E −△E = 0 , (3–1)

50

Figure 3-1. Panel (a): Double array of dielectric cylinders. The unit of length is the arrayperiod. The axis of each cylinder is parallel to the y-axis and is at a distanceh from the x-axis.Panel (b): The energy spectrum for a Schrodinger equation with radiallysymmetric potential consists of a discrete spectrum of bound states withnegative energies and the radiation continuum. The latter may containadditional bound states; these are the bound states in the radiationcontinuum.Panels (c) and (d): Harmonically time dependent solutions to Eq.(3–1) arelabeled by points of the spectral cylinder as explained in text. The diffractionthresholds E±n are rings that partition the cylinder into sectionscorresponding to a fixed number of open channels. In particular, E0 is thethreshold for the radiation continuum below which there can be no scatteringstates.

where the dielectric function ε is piecewise constant; it is equal to εc > 1 on the

scatterers and to 1 otherwise. For a harmonic time dependent electric field E(r, t) =

Eω(r)e−iωt , the amplitude Eω(r) satisfies the equation

[−△+ k2(1− ε)]Eω = k2Eω , (3–2)

where k2 = ω2/c2 is the spectral parameter. This equation is similar to the Schrodinger

equation with an attractive (negative) potential 1 − ε. If k is the wave vector of the

incident radiation, then the periodicity of the structure requires that solutions to Eq.(3–2)

satisfy Bloch’s theorem,

Eω(r + e1) = e ikxEω(r) , (3–3)

51

where k = kxe1 + kze3 and {e1, e2, e3} are the unit vectors along the coordinate axes.

Consequently, the wave vectors of the scattered radiation can only have the following

form

k±m = (kx + 2πm)e1 ±

√k2 − (kx + 2πm)2e3 , m = ...− 2,−1, 0, 1, 2... (3–4)

with the convention√k2 − (kx + 2πm)2 = i

√(kx + 2πm)2 − k2 if k2 < (kx + 2πm)2. The

values of m for which k2 ≥ (kx + 2πm)2 label open diffraction channels, all other values

of m correspond to closed channels. If, for instance, the incident radiation propagates in

the positive z−direction as shown in Fig. 3-1(a), then the transmitted field reads

Eω(r) = e ik·r +∑m

Tmeik+m·r , z > h + R , (3–5a)

where Tm are the amplitudes of the transmitted modes and the first term describes the

incident radiation whose amplitude is set to one. Similarly, in the region to the left of the

structure, the electric field is given by

Eω(r) = e ik·r +∑m

Rmeik−m ·r , z < −h − R , (3–5b)

where Rm are the amplitudes of the reflected modes. The choice between k+m and k−m

is dictated by the outgoing wave boundary condition at the spatial infinity |z | → ∞. In

particular, the field corresponding to closed channels in Eqs.(3–5) decays exponentially

in the asymptotic region |z | → ∞. So the closed channels do not contribute to the

energy flux carried by the scattered wave. The scattered flux is carried only by the

radiation in open diffraction channels. The condition k2m = (kx + 2πm)2 defines the

threshold for opening the mth diffraction channel. The coefficients Rm and Tm for open

channels are the reflection and transmission coefficients, respectively.

52

The transmission and reflection amplitudes may be inferred from the solution Eω of

the Lippmann-Schwinger integral equation [10]

Eω(r) = e ik·r +k2

4π

∫(ε(r0)− 1)Eω(r0)G(r|r0)dr0 , (3–6)

in which G(r|r0) = iπH0(k |r − r0|) is the 2-dimensional free-space Green’s function for

the Helmholtz operator △ + k2 with outgoing boundary conditions, and H0 is the Hankel

function of the first kind of order 0.

Observe that, unless there is no incident radiation (the term e ik·r is omitted), the

solutions to Eq.(3–2) cannot be square integrable along the z-axis. By the analogy

with quantum scattering theory for radially symmetric potentials, such solutions are

in the radiation continuum of the energy spectrum. In quantum theory, solutions to

the Schrodinger equation are eigenvectors of the energy operator whose spectrum E

form a subset in the real line. The upper part (positive) of the spectrum is continuous

and corresponds to scattered states that can carry the probability flux to the spatial

infinity. Below the continuous part of the spectrum, i.e., E < 0, the energy spectrum

is discrete (Fig. 3-1(b)). It corresponds to bound states. Bound states have a finite L2

norm (they decay fast enough at the spatial infinity). It was first proved by Von Neumann

and Wigner in 1929 that under special circumstances, there might exist bound states

in the radiation continuum. A counterintuitive physical peculiarity of such states is that

a bound state is a standing wave in a potential well (an attractive potential), while the

conventional quantum mechanical wisdom would suggest that for a potential bounded

above a standing wave with the energy in the radiation continuum should tunnel through

the potential barrier to the spatial infinity and, hence, cannot be stable. Nevertheless,

such states do exist and the theory of their formation is now well developed [6, 8, 12].

The goal here is to establish a similar picture for electromagnetic excitations in the

periodic double array of dielectric cylinders, and, specifically, to find the conditions on

the physical parameters of this system under which the bound states in the radiation

53

continuum exist, their eigen-frequencies, and the analytic form of the corresponding

electromagnetic fields. The very existence of bound states for this system was first

demonstrated by numerical simulations [11]. Here a complete analytic study of the

system is given.

Any bound state is a solution of the generalized eigenvalue problem (3–2) (no

incident radiation term) and, hence, is fully characterized by the spectral parameter

E = k2 > 0 and the Bloch phase factor e ikx because of the boundary condition (3–3).

The pair (E , e ikx ) is viewed as a point on the (half) cylinder R+ × S1 where E ∈ R+ and

e ikx ∈ S1. It will be called a spectral cylinder (or spectral space) of a periodic grating.

In contrast to quantum mechanical bound states in spherically symmetric systems, the

Bloch boundary condition requires a more adequate classification of bound states here.

In order to identify bound states in the radiation continuum, one has first to determine

the region of the spectral cylinder occupied by the radiation states. In the asymptotic

region |z | → ∞, a harmonic time dependent solution E = Eωe−iωt to Eq.(3–1) is

characterized by the pair (E , kx) where E = k2 = ω2/c2. Given E and kx , the field outside

the scattering region is completely determined by its behavior in the diffraction channels

as specified in Eqs.(3–5). Consequently, if two radiation modes have the same E , while

their parameters kx differ by a 2π-multiple, say, by 2πm0, then they have exactly the

same open diffraction channels because in the classification introduced in Eq.(3–4) the

difference of channels would merely mean the relabeling m → m − m0 (or the same

change of the summation index in Eq. (3–5)). Therefore the radiation modes correspond

to points (E , e ikx ) on the spectral cylinder for which one or more diffraction channels are

open.

The spectral cylinder can be partitioned into sections associated with a fixed

number of open diffraction channels. The diffraction thresholds appear as curves

separating these portions of the cylinder. Indeed, let [kx ] designate the argument of

e ikx in (−π,π], i.e., [kx ] = kx mod 2π. Then, up to the aforementioned reordering, the

54

diffraction thresholds on the spectral cylinder are exactly

E±n(kx) = (2πn ± |[kx ]|)2 , n = 0, 1, 2, 3... , (3–7a)

and they appear in the order,

E0 ≤ E−1 ≤ E1 ≤ E−2 ≤ E2 ≤ E−3 ≤ E3 ... (3–7b)

If kx is identified with the angular variable spanning the compactified direction of the

spectral cylinder, then the diffraction thresholds are curves in an ever rising order on the

spectral cylinder with nodes on the axes [kx ] = 0 and [kx ] = π. The curve E = E0(kx)

is the threshold below which no radiation modes exist, and therefore, the radiation

continuum lies immediately above this curve. This continuum is split into distinct regions

corresponding to a fixed number of open channels by consecutive thresholds as

indicated by Eq.(3–7b). These regions will be labeled as radiation continuum I, radiation

continuum II, radiation continuum III, etc., where the Roman numeral indicates the

number of open channels in each region (Fig. 3-1(b), (c), and (d)).

All the solutions to Eq.(3–2) below the threshold E0, if any, must be bound states,

i.e., they decay exponentially in all diffraction channels and, hence, have a finite L2 norm

in the space S1 × R spanned by (x , z) (x is compactified into a circle S1 because of the

boundary condition (3–3)). In contrast, radiation modes behave as harmonic functions in

the asymptotic region |z | → ∞ and, hence, do not have a finite L2 norm. So the problem

is to find, if any, square integrable solutions on S1 × R above the curve E = E0 which are

the sought-for bound states in the radiation continuum.

3.2 Bound States

As defined above, bound states are square integrable solutions of the homogeneous

Lippmann-Schwinger integral equation

Eω(r) =k2

4π

∫(ε(r0)− 1)Eω(r0)G(r|r0)dr0 , k2 > 0 , (3–8)

55

which satisfy Bloch’s boundary condition (3–3). The square integrability here means a

finite L2 norm in S1 × R (with the x-direction compactified into a circle). As also noted

above, Eq.(3–7) is understood in the distributional sense (Appendix A.1). This is a

generalized eigenvalue problem because the Green’s function G(r|r0) also depends on

the spectral parameter k2. In this problem, the Bloch parameter kx may be restricted

to the interval [−π,π]. It must be stressed that this restriction is only permitted by the

absence of an incident wave which otherwise determines the phase factor in (3–3).

The task is to determine the values of a, kx , h and k that allow for the existence of

nontrivial solutions Eω to Eq.(3–8) for fixed radius R and fixed dielectric constant εc in

the limit of thin cylinders, i.e., kR ≪ 1. In what follows, the existence of a bound state

always means the existence of a wavenumber k at which the bound state Eω occurs.

The main results established in the present study may be split into the following cases

which are also summarized in Fig. 3-2(a):

• Below the radiation continuum: Bound states exist for all kx , a and for all distancesbetween the arrays.

• Continuum I (one open diffraction channel): Bound states only exist if

– either kx = 0 and a ∈ [0, 12 ] is arbitrary

– or a ∈ {0, 12} and kx ∈ (−π,π) is arbitrary

Under these conditions, for each pair (a, kx) there is a discrete set of distancesbetween the arrays at which bound states exist.

• Continuum II (two open diffraction channels): Bound states only exist fora = 0 or a = 1

2and for a certain dense set of values of kx .

For each allowable pair (a, kx), there is exactly one or two distances between thearrays at which the bound states exist.

• Continuum N, N≥ 3 (three or more open diffraction channels): Bound states existonly for specific values of the radius R and the dielectric constant εc .

In all the above cases, bound states occur in two types, symmetric and skew-symmetric

relative to the transformation z → −z and x → a + x , when the shift parameter takes the

56

Figure 3-2. Panel (a): Values of the parameters a and kx for which bound states exist.The bottom level represents the modes below the radiation continuum, thesecond level represents the modes in the radiation continuum I, and the thirdlevel represents the modes in the radiation continuum II.Panels (b) and (c): The shaded areas represents the region of validity ofinequalities (3–29) and (3–39), respectively.

boundary values, a = 0 or a = 12 . However, this classification cannot be established for

double arrays with intermediate values of the shift parameters, a ∈ (0, 12).

The above classification of the bound states in the cases a = 0 and a = 12 is relevant

when analyzing the (discrete) values of the distance 2h between the two arrays at which

the bound states occur. When the double array is symmetric (a = 0), then these values

always exist in the whole range of h ∈ (R,∞). On the contrary, skew-symmetric bound

states will only occur for those values of h that exceed a certain minimal threshold, and

this threshold increases as R2(εc − 1) → 0. In other words, when the scattering structure

becomes more transparent, the two arrays have to be taken further apart in order

for skew-symmetric bound states to form. A similar phenomenon is observed for the

skew-symmetric array (a = 12). In this case, however, the skew-symmetric modes behave

as the symmetric ones in the previous case and vice versa, i.e., the minimal threshold

distance exists for the symmetric modes and increases as the system becomes more

transparent (R2(εc − 1) → 0), while the skew-symmetric modes occur at a discrete set of

values of h in the whole range h ∈ (R,∞).

The existence of the above classification can be established by studying the

symmetry of the function ε(x , z). Consider the operator Pa defined by PaEω(x , z) =

e−iakxEω(x + a,−z). For a = 0 or a = 12 , the operator Pa commutes with the operator H

57

of Eq.(3–2) and has eigenvalues ±1 since it is a projection, i.e., P2a = 1. By virtue of the

commutativity, each bound state Eω is an eigenfunction of Pa and, hence,

Eω(x + a,−z) = ±e iakxEω(x , z) .

Symmetric states may then be defined as those for which Eω(x + a,−z) = e iakxEω(x , z)

while the skew-symmetric states satisfy Eω(x + a,−z) = −e iakxEω(x , z). For a = 0 this

simply means that the first states are even in z while the second are odd.

This description cannot be generalized to an arbitrary shift a since Pa no longer

commutes with H if a ∈ (0, 12). Instead, it follows from the symmetry of ε(x , z) that

the operator QaEω(x , z) = Eω(a − x ,−z) commutes with H for all a and also Q2a = 1.

However, the operator Qa is not symmetric in the subspace of functions in L2(S1 × R)

satisfying condition (3–3) unless kx = 0 or kx = ±π. Indeed if it were, then any bound

state Eω would have been an eigenfunction of Qa, and, therefore, Eω(a − x ,−z) =

±Eω(x , z). By replacing x by x + 1 and applying condition (3–3) one gets e ikxEω(x , z) =

e−ikxEω(x , z) so that kx = 0 or kx = ±π.

As noted above, the bound states are found by perturbation theory in the limit of

thin cylinders. The technical details are given in Appendix A.2 where it is shown that

non-trivial solutions to Eq.(3–7) can only exist if the fields Eω(−he3) and Eω(ae1 + he3)

on the two cylinders positioned at (0,−h) and (a, h) respectively do not simultaneously

vanish and satisfy the homogeneous system of equations�0Eω(ae1 + he3) + �+Eω(−he3) = 0

�−Eω(ae1 + he3) + �0Eω(−he3) = 0

(3–9)

in which the coefficients �0 and �± are given by the following expressions

�0(k , kx) =

∞∑m=−∞

(1

kz ,m− 1

2πi(|m|+ 1)

)+

i

2π

(1

δ0(k)+ 2 ln(2πR)

)

�±(a, h, k , kx) =

∞∑m=−∞

e i(±a(kx+2πm)+2hkz ,m)

kz ,m(3–10)

58

with kz ,m =√k2 − (kx + 2πm)2 being the z-component of the wave vector in the mth

diffraction channel and

δ0(k) =

(kR

2

)2

(εc − 1) (3–11)

In particular δ0(k) > 0 as εc > 1. It should also be kept in mind throughout the work

that δ0(k) ≪ 1 in the limit considered kR ≪ 1. In terms of the fields on the cylinders in

Eqs.(3–9), the electric field strength everywhere off the scatterers is then given by,

Eω(r) = 2πiδ0(k)

(Eω(ae1 + he3)

∑m

e i((x−a)(kx+2πm)+|z−h|kz ,m)

kz ,m

+ Eω(−he3)∑m

e i(x(kx+2πm)+|z+h|kz ,m)

kz ,m

) (3–12)

The system of Eqs.(3–9) admits non-trivial solutions if and only if its determinant

�(a, h, k , kx) = �20−�+�− (3–13)

vanishes at some point (a, h, k , kx) in the space of system parameters. This is the

condition for bound states to exist, no matter if they are below or in the radiation

continuum.

In the following two subsections, roots of �(a, h, k , kx) in k for fixed a,h, and kx are

analyzed to find bound states below the radiation continuum as well as bound states

in the continuum I. The analysis of the higher continua in the spectrum, while being

similar to the case of the continuum I, is technically more involved. To avoid excessive

technicalities before the discussion of applications of resonances with the vanishing

width to a near field amplification, the analysis of higher continua is postponed to

Section 3.3. In each study, the spectral parameter k ranges over an open interval in

which the functions �0 and �± are analytic in k and diverge at the endpoints. Also,

note that, since �(a, h, k , kx) is even in kx , the range of this parameter can be reduced

from [−π,π] to [0,π]. Throughout the rest of this section as well as in Section 3.3, the

59

symbols �∗, �c , and �s denote, respectively, the following functions:

�c(a, h, k , kx) =∑mcl

e−2hqz ,m

qz ,mcos(2πam) , �s(a, h, k , kx) =

∑mcl

e−2hqz ,m

qz ,msin(2πam)

�∗(k , kx) = Im (�0(k , kx)) (3–14)

where the superscript ”cl” in mcl means that the sums are taken over all m’s that

correspond to closed diffraction channels, and qz ,m is the imaginary part of kz ,m when

the mth channel is closed, i.e., qz ,m =√(kx + 2πm)2 − k2 if k2 < (kx + 2πm)2. Recall that

kz ,m =√k2 − (kx + 2πm)2. By construction, the functions (3–14) are always real-valued.

3.2.1 Bound States Below the Radiation Continuum

Bound states below the radiation continuum are nontrivial solutions to the

homogeneous Lippmann-Schwinger integral equation when all diffraction channels

are closed, kx ∈ (0,π] and 0 < k < kx . In this case the determinant of Eq.(3–13)

factorizes as

�(a, h, k , kx) = −+−, ± = �∗ ∓√�2

c +�2s (3–15)

hence bound states will exist for wavenumbers k at which the functions + and −

vanish. That such wavenumbers exist for + follows from the limits

limk→0+

+(k) = +∞ , limk→kx

−+(k) = −∞

and the Intermediate Value Theorem. In particular, for each triplet (a, h, kx), there exists

a wavenumber k+(a, h, kx) at which +(k+) = 0. It is shown in Appendix A.4 that in the

leading order of δ0(kx),

k+ ≈ kx −8π2δ2

0(kx)

kx. (3–16)

The function − can also be shown to have roots when the distance 2h is

sufficiently large or when kx is close to π, this gives a second family of wavenumbers

k−(a, h, kx) at which bound states occur. These assertions result from the limits at 0 and

60

at kx of −. These limits are,

limk→0+

−(k) = ∞ ,

limk→kx

−−(k) = −4h cos2(πa)− sin2(πa)√

π(π − kx)+

1

2πδ0(kx)+O(1) .

In particular for fixed kx ∈ (0,π] and a = 12 , h can be chosen sufficiently large for the

last limit to be negative and therefore by the Intermediate Value Theorem a wavenumber

k− such that −(k−) = 0 exists on the interval (0, kx). Conversely, if h is fixed, then

limkx→π−

limk→kx

−− =

−∞ if a = 0

−4h +1

2πδ0(π)+O(1) if a = 0

(3–17)

Thus if 0 < a ≤ 12 and kx is sufficiently close to π then the equation −(k) = 0 has a root

k−(a, h, kx). When a = 0, the sign of the limit in Eq.(3–17) depends on the radius of the

scattering cylinders, the dielectric constant εc and the distance between the two arrays.

If h is large enough or R2(εc − 1) is not too small, then the limit is negative and the

existence of the wavenumber k− follows. Because of a complicated dependence of the

existence condition for the second family of wavenumbers on the physical parameters of

the system, such a simple analytic approximation as for k+ above is not possible for k−.

Finally, from Eqs.(3–9) and (3−−12) the bound state solutions E± at the

wavenumbers k± are obtained. As any solution to a homogeneous equation, they

can only be determined up to a normalization constant which is chosen to be the value

of the electric field E±(−he3) on the cylinder at (0,−h). In terms of this value, the

electric field on the cylinder at (a, h) is then

E±(ae1 + he3) = ±e i(ϕ+akx )E±(−he3), ϕ = arg

(∑m

e−2hqz ,m+2πiam

qz ,m

)∣∣∣∣∣k=k±

and everywhere off the scattering cylinders it is:

E±(r) = 2πiδ0(k)E±(−he3)

∑m

e ix(kx+2πm)

kz ,m

(e i |z+h|kz ,m ± e i(|z−h|kz ,m−2πam+ϕ)

) ∣∣∣∣∣k=k±

(3–18)

61

3.2.2 Bound States in the Radiation Continuum I: One Open Diffraction Channel

When only the 0th-order diffraction channel is open, kx ∈ [0, π) and kx < k < 2π − kx ,

the determinant of Eq.(3–13) can be rewritten in the following form convenient for the

analysis

�(a, h, k , kx) =sin2(2hkz)

k2z+�2

s −+− +2i

kz

(+ sin2(hkz) + − cos2(hkz)

)(3–19)

where ±(a, h, k , kx) = �∗(k , kx) ±(sin(2hkz )

kz−�c(a, h, k , kx)

)and the functions �∗,�c

and �s are given in Eqs.(3–14). Thus, bound state exist if both the real and imaginary

parts of (3–19) vanish: sin2(2hkz)

k2z+�2

s = +−

+ sin2(hkz) + − cos2(hkz) = 0

(3–20)

The first of these equations implies that +− ≥ 0. If this inequality were to be strict,

then the second equation would not have held, and, therefore, +− = 0. Thus, the first

equation implies that �s = 0 and sin(2hkz) = 0. In turn, the latter equation implies that

either cos(hkz) = 0 or sin(hkz) = 0, and, therefore, the system of Eqs.(3–20) splits into

two systems, namely,�s(a, h, k , kx) = 0

cos(hkz) = 0

+(a, h, k , kx) = 0

�s(a, h, k , kx) = 0

sin(hkz) = 0

−(a, h, k , kx) = 0

(3–21)

To solve the first equation of each system, the series for �s is rewritten as,

�s = −∞∑

m=1

cm sin(2πam), cm =e−2hqz ,−m

qz ,−m

− e−2hqz ,m

qz ,m(3–22)

Recall that qz ,m =√(kx + 2πm)2 − k2. In particular, if kx = 0 then cm = 0, ∀m = 1, 2, 3...

and the equation �s = 0 holds trivially. Similarly, if a = 0 or a = 12 , then the equation

62

holds trivially as sin(2πam) = 0, ∀m. It turns out that these are the only possible roots of

�s if h > ln 24π

≈ 0.055. This conclusion stems from the following factorization of �s ,

�s(a, h, k , kx) = − sin(2πa)

∞∑m=1

[(cm − 2cm+1 + 2

∞∑n=1

(cm+2n − cm+2n+1)

)sin2(πam)

sin2(πa)

](3–23)

together with the facts that for kx = 0,

cm > 0 and∞supm=1

{cm+1

cm

}= e−4πh (3–24)

These statements are established in Appendix A.3. When h > ln 24π

, then cm+1 <12cm,∀m

and therefore each of the summands in the series of Eq.(3–23) is nonnegative. Since

the first term of the said series does not vanish, it follows that the series does not vanish.

Consequently, if kx = 0 then �s = 0 if and only if sin(2πa) = 0 ,i.e., a = 0 or a = 12 .

It is remarkable that under the restriction h > ln 24π

no solution to Eqs.(3–21) is lost.

Indeed, since sin(2hkz) = 0 at a bound state, it follows that

h =nπ

2kz, kz =

√k2 − k2x

for some positive integer n. In the ranges considered, kz < 2π and therefore h > n4≥

14> ln 2

4π. Thus a necessary condition for the existence of bound states is that either

kx = 0 while a ∈ [0, 12 ] is arbitrary or a is either 0 or 12 while kx ∈ [0, π) is arbitrary. In set

notation,

(a, kx) ∈ L =([0, 12 ]× {0}

)∪({0, 12} × [0, π)

)(3–25)

The set L is represented by the second level of Fig.3-2(a).

63

Let us turn to solving the last two equations in each of the systems in Eqs.(3–21).

For this purpose, the function n is defined for each positive integer n by,

n(k , kx , a) =

+(a,

nπ

2kz, k , kx) if n is odd

−(a,nπ

2kz, k , kx) if n is even

=1

2π

(1

δ0(k)+ 1 + 2 ln(2πR)

)

+∑m =0

(1

2π(|m|+ 1)− 1− (−1)n cos(2πam)e−nπqz ,mk

−1z

qz ,m

)(3–26)

where k ∈ (kx , 2π − kx). Then the systems of Eqs.(3–21) split into the countable set of

systems h =

nπ

2√k2 − k2x

n(k , kx , a) = 0

n = 1, 2, 3... (3–27)

where the systems corresponding to odd n arise from the first of systems (3–21) and

those corresponding to even n result from the second system. In each of the systems

(3–27), the second equation determines the wavenumbers at which bound states occur.

In turn, by the first equation, these wavenumbers determine the distances h that allow

for the bound states to exist. In the next paragraph it is shown that, for fixed (a, kx) ∈ L,

each system can admit at most one solution. Hence the set of distances h allowing the

existence of bound states is discrete.

Let (a, kx) ∈ L be fixed. It is shown in Appendix A.3 that the function k 7→ n is

monotone decreasing on its domain (kx , 2π− kx) and therefore admits atmost one root in

the said domain. Moreover, the root only exists if the limits of k 7→ n at kx and 2π − kx

are of opposite sign. Specifically, the limit at kx must be positive while the limit at 2π − kx

64

must be negative. The first limit is,

limk→k+x

n =

∞ if kx = 0

1

2π

(1

δ0(kx)+ 1 + 2 ln(2πR)

)+∑m =0

(1

2π(|m|+ 1)− 1√

4π2m2 + 4πmkx

)if kx = 0

The requirement that this limit be positive when kx = 0 puts a restriction on the values of

R and εc that allow for the existence of bound states. However, this is a too complicated

condition to analyze. A weaker, but easier to analyze, condition is obtained by first

rewriting the positivity condition as,

2

πR2(εc − 1)k2x>

∞∑m=1

(1√

4π2m2 − 4πmkx+

1√4π2m2 + 4πmkx

− 1

πm

)+1

π

(1

2− ln(2πR)

)(3–28)

The rearrangement of the series is made to ensure that all the summands in the series

are nonnegative; they vanish at kx = 0. Also, since the cylinders are thin, it may be

assumed that R <√e

2π≈ 0.262 so that 1

2− ln(2πR) > 0. Therefore all the summands

in Eq.(3–28) are nonnegative and hence the left hand side must be larger than each

of the summands on the right individually. To estimate the threshold value of R√εc − 1

below which bound states may exist, the first term (m = 1) is retained in the sum (3–28).

This term is then written in the form k2x (π − kx)−1/2g(kx) to isolate its branch point at

kx = π and its root at kx = 0 which is of multiplicity 2. The minimization of g(kx) on [0, π]

produces an estimate:

R√εc − 1 <

C 4√π − kx

k2x(3–29)

where

C = π34

√2

(min0≤t≤1

1 +√1 + t +

√1− t√

1 + t(1 +

√1− t2

) (2 +

√1 + t +

√1− t

))− 12

≈ 5.846

65

to observe In particular when kx is close to π, the quantity R√εc − 1 must be small

enough in order for bound states to exist at all. Figure 3-2(b) shows the regions in which

Eq.(3–29) is valid.

As specified above, for bound states to exist it is required that the limit of n at

2π − kx be negative. This limit is,

limk→(2π−kx )−

n =

−∞ if n is odd and a = 1

2or n is even and a = 0

− nπ√π(π − kx)

+1

2πδ0(2π − kx)+O(1) otherwise

In particular, the limit is negative except possibly when n is odd and a = 12 or n is even

and a = 0. Even in the latter cases however, the negativity condition may be ensured

by taking kx sufficiently close to π so that k−1z is large or by choosing n sufficiently large.

Thus, if the parameters R and εc of the scattering cylinders verify condition (3–28),

bound states in the continuum I do exist. To be precise, given a positive integer n; then

• ∀(a, kx) ∈((0, 1

2) × {0}

)∪({0} × [0, π)

)there exists a bound state at the

wavenumber k2n−1(a, kx) ∈ (kx , 2π − kx) and at the distance 2h2n−1(a, kx) =(2n−1)π√k22n−1−k2x

between the two arrays of cylinders. When a = 12 , then the wavenumber

k2n−1 exists for sufficiently large n or for kx sufficiently close to π.

• ∀(a, kx) ∈((0, 1

2) × {0}

)∪({12} × (0,π)

)there exists a bound state at the

wavenumber k2n(a, kx) ∈ (kx , 2π − kx) and at the distance 2h2n(a, kx) = 2nπ√k22n−k2x

between the two arrays of cylinders. When a = 0, then the wavenumber k2n existsfor sufficiently large n or for kx sufficiently close to π.

In the limit of the thin cylinders considered, approximate values can be inferred for

the wavenumbers kn, n = 1, 2, 3.., by only keeping the leading terms in the equations

n(k) = 0. This is detailed in Appendix A.4. For instance if n is odd and a = 0 or n is

even and a = 12 , then the wavenumber kn(kx) and the distance hn(kx) are approximated

66

in the leading order of δ0 by,

kn(kx) ≈ 2π − kx −8π2δ2

0(2π − kx)

2π − kxhn(kx) ≈

nπ

4√π(π − kx)

(1 +

2πδ20(2π − kx)

π − kx

)(3–30)

Finally, from Eqs.(3–9) and (3–12) the explicit form the electric field for the bound

states {En}∞n=1 is obtained at the wavenumbers {kn}∞n=1 and the distances {2hn}∞n=1

between the arrays of cylinders. The eigenfunction of bound states in the continuum can

only be determined up to a multiplicative constant which is chosen to be the value of the

electric field En(−hne3) on the cylinder at (0,−hn). In terms of this value, the electric field

on the cylinder at (a, hn) is then,

En(ae1 + hne3) = (−1)n+1e iakxEn(−hne3)

and everywhere off the scattering cylinders it is:

En(r) = 2πiδ0(kn)En(−hne3)∑m

e ix(kx+2πm)

knz ,m

(e i |z+hn|knz ,m + (−1)n+1e i(|z−hn|knz ,m−2πam)

)(3–31)

where knz ,m =√k2n − (kx + 2πm)2. It can be verified easily that outside the scattering

region, i.e., |z | > hn, there is no contribution to the field En from the 0−order diffraction

channel. With this channel being the only open channel, it follows that En decays

exponentially in the asymptotic region |z | → ∞ and therefore it is square integrable on

S1 × R as required. Figure 3-3 shows examples of plots of the absolute values of the

fields En.

3.2.3 Application: Zero Width Resonances and Near Field Amplification

The study of Section 3.2 shows that, given two parallel arrays of subwavelength

dielectric cylinders of radius R and dielectric constant εc , there are points (a, kx) for

which a bound state exists if the distance between the arrays attains a specific value

2hb; this value also determines the wavenumber kb of the bound state. Throughout the

following discussion, the pair (a, kx) is fixed.

67

Figure 3-3. The modes En as defined in Eq.(3–31) for a = 0, R = 0.1, εc = 1.5, andkx = 0. The panels show En as a function of x (vertical axis) and z

(horizontal axis). The color shows the absolute value of En as indicated onthe left inset. The positions of cylinders are indicated by a solid black curve.The values of hn are shown below each panel. Top left: E1, the symmetricmode for the smallest h = h1. Top right: E2, the skew-symmetric mode ath = h2. Bottom left: E3, the second symmetric mode at h = h3. Bottom right:E4, the second skew-symmetric mode at h = h4.

Consider the scattering problem for the double array when h = hb. If the incident

wave has the wavenumber kb, then the solution to Eq.(3–2) is not unique as any solution

of the homogeneous equation can always be added. The latter are the bound states. Of

course, this ambiguity is related to the fact that the incident radiation cannot excite the

bound state (which is a wave-guiding mode propagating along the array), and, hence, an

additional condition must be imposed that the bound state is not present in the system

at the very beginning if one wishes to have a unique solution. On the other hand, the

presence of a bound state has no effect whatsoever on the scattering amplitudes as

they are defined in the far field zone (z → ±∞) to which the bound state gives no

contribution anyway. This degeneracy disappears as soon as (h, k) = (hb, kb). In the

latter case, the solution Eω to Maxwell’s equations is uniquely determined by Eq.(3–6).

Such irregularity suggests that, as a function of h and k , the field Eω is not analytic in the

vicinity of the points (hb, kb). This is indeed the case. It is shown in what follows shortly

that the values of the reflected flux as well as those of the fields inside the cylinders near

the points (hb, kb) depend on the path along which these points are approached in the

h, k−plane. From a mathematical point view, this lack of analyticity is explained by the

68

presence of simple poles at the wavenumbers kb in the field Eω when considered as a

function of k for fixed h = hb. The objective here is to exploit the existence of these poles

to show that the evanescent field in the scattering problem is amplified as compared

to the amplitude of the incident field when (h, k) is close to (hb, kb) in some regions of

the array, in particular, on the cylinders. The effect can therefore be used to amplify

optical non-linear effects in the structure in a controllable way. This is illustrated with an

example of one open channel for an array without the shift, i.e., a = 0.

Suppose that a plane wave of wavenumber k ∈ (kx , 2π − kx) and unit amplitude

impinges the double array. In this case, the specular reflection coefficient which is the

ratio of the reflected flux to the incident flux at the spatial infinity is,

R = |R0|2

where R0 is the reflection coefficient of the only open diffraction channel, namely, the 0

order channel as given in Eq.(3–5b). If (h, k) is not one of the points (hb, kb), then the

reflection coefficient R0 and the fields inside the cylinders are,

R0 = − cos2(hkz)

cos2(hkz) +12ikz+

+sin2(hkz)

sin2(hkz) +12ikz−

(3–32a)

Eω(±he3) =ikz

2πδ0(k)

(cos(hkz)

cos2(hkz) +12ikz+

± isin(hkz)

sin2(hkz) +12ikz−

)(3–32b)

for the functions ± of Eq.(3–19) (Appendix A.2). The denominators in both expressions

are factors of the determinant �(0, h, kx , k)in Eq.(3–19) (Here a = 0). In particular, the

points (hb, kb) are roots of the denominators to the specular coefficient and the fields

inside the cylinders.

For the illustration purpose, the behavior of the specular coefficient and the

fields inside the cylinders are studied as (h, k) approaches a critical point (hb, kb) =

(h2n−1, k2n−1) for some positive integer n in the h, k−plane. As described in Subsection 3.2.2;

69

at the point (h2n−1, k2n−1) the following system holds,cos(hkz) = 0

+(h, k) = 0

In the rest, the curves cos(hkz) = 0 and +(h, k) = 0 will be denoted by Cc and C+

respectively and their intersection points, i.e., the points (h2n−1, k2n−1), will be denoted by

P2n−1.

The first observation is that as (h, k) approaches P2n−1, then cos(hkz) → 0 and

+(h, k) → 0 independently as the curves Cc and C+ intersect at a nonzero angle at

P2n−1. This may be established through the linearizations of the functions (h, k) 7→

cos(hkz) and (h, k) 7→ kz+(h, k) at (h2n−1, k2n−1). If �h = h − h2n−1 and �k = k − k2n−1,

then in the vicinity of P2n−1,

cos(hkz) ≈ ξ(�h, �k) = (−1)n(h2n−1k2n−1

kz ,2n−1

�k + kz ,2n−1�h

)1

2kz

+(h, k) ≈ η(�h, �k) =1

2kz ,2n−1

(∂k

+(h2n−1, k2n−1)�k + ∂h+(h2n−1, k2n−1)�h

)where kz ,2n−1 =

√k22n−1 − k2x . The functions ξ and η are then linearly independent if,

h2n−1k2n−1

kz ,2n−1

∂h+(h2n−1, k2n−1)− kz ,2n−1∂k

+(h2n−1, k2n−1) = 0

That this condition indeed holds can be proved by examining the function 2n−1(k) =

+((2n−1)π

2kz, k)

introduced in Eq.(3–26). In Appendix A.3 it is shown that ∂k2n−1(k) < 0

for all k ∈ (kx , 2π − kx). Consequently,

∂k2n−1(k2n−1) = −h2n−1k2n−1

k2z ,2n−1

∂h+(h2n−1, k2n−1) + ∂k

+(h2n−1, k2n−1) < 0

This establishes the linear independence of ξ and η. Thus, as (�h, �k) → (0, 0), there

should be ξ → 0 and η → 0 independently. In the vicinity of the critical point P2n−1 the

70

principal parts of R0 and Eω(±he3) are then,

R0(h, k) ≈1

1 + ikz ,2n−1−(h2n−1, k2n−1)+

ξ2(�h, �k)

ξ2(�h, �k) + iη(�h, �k)(3–33a)

Eω(±he3) ≈ikz ,2n−1

2πδ0(k2n−1)

(±i (−1)n+1

1 + ikz ,2n−1−(h2n−1, k2n−1)+

ξ(�h, �k)

ξ2(�h, �k) + iη(�h, �k)

)(3–33b)

The first summands in each of these equations are constant and obey the estimate,

1

1 + ikz ,2n−1−(h2n−1, k2n−1)∼ δ0(k2n−1)

kz ,2n−1

The second summands in Eqs.(3–33) account for the lack of analyticity of the specular

coefficient R and the fields Eω(±he3) in the vicinity of the critical point P2n−1. In

particular, since η ≡ 0 along the tangent line to C+ at P2n−1, it follows that along this

tangent line and hence along the curve C+,

Eω(±he3) ≈(−1)ni

2πδ0(k22n−1)(1− h2n−1k2n−1

k2z ,2n−1

∂h+

∂k−

)�h

, �h = h − h2n−1 → 0

Thus the electric field inside the cylinders diverges at the points P2n−1.

For the specular coefficient, the significance of the points P2n−1 is that they are

positions of resonances with the vanishing width, which, in turn, demonstrates that

the bound state in the radiation continuum are interpreted as resonances with the

vanishing width in the formal scattering theory. Indeed, if h = h2n−1 is fixed and kr(h)

is a wavenumber such that +(h, kr(h)) = 0, then kr is a resonant wavenumber for the

specular coefficient R. For k near kr , the Breit-Wigner theory asserts that R0 will have

the form,

R0 ∼i�

k − kr + i�

where 2� is the resonance width of the Lorentzian profile of R = |R0|2. The half-width �

may be found by expanding the function k 7→ +(h, k) in a Taylor series at the resonant

71

wavenumber kr , it is,

� = −2 cos2(hkz)

kz∂k+

∣∣∣∣∣k=kr

so that at the points P2n−1 this width vanishes. Figure 3-4 shows plots of the specular

coefficients and the electric field along the curve C+.

Figure 3-4. The specular coefficient and the electric field on the cylinders near a boundstate in the continuum for a = 0, R = 0.1, and εc = 1.5.Panel (a): Shows the specular reflection coefficient as a function of h and thewavenumber k . It is plotted for kx = π

5near the threshold k−1 =

9π5

. Thespecular coefficient is very close to its maximum along the curves±(h, k) = 0 that determine the resonance positions. The lower (upper)bright region roughly corresponds to the curve +(h, k) = 0 (−(h, k) = 0).The bound states correspond to the points separating two consecutive brightstrips. At the bound states the specular coefficient lacks analyticity and itsvalue depends strongly on how the bound state is approached in the(h, k)-plane.Panel (b): The specular coefficient R(k(h), h) (dashed blue curve) and theabsolute value of the electric field Eω(he3)(k(h), h) (solid red curve) on thecylinder at (x , z) = (0, h) of the double array where k = k(h) is implicitlydefined by +(h, k) = 0. Along this curve, the electric field diverges near thebound states.Panel (c): Same legend as for Panel (b) in the case of two open channelsand kx = π. In this case too, the fields on the cylinders diverge in the vicinityof the bound states.

3.3 Bound States in the Radiation Continuum N, N≥ 2

When more than one diffraction channel are open, the bound states may still be

shown to exist. However they become rarer as the number of diffraction channels

72

increases. The physical reason for that is simple. As suggested in the introduction, a

bound state is formed due to a destructive interference of the decay radiation of two

quasi-stationary electromagnetic modes localized in the vicinity of each array. If more

than one decay channel are open for these modes, then the destructive interference

must occur in all the decay channels in order for a bound state to form, which puts

more restrictions on the system parameters. Indeed, consider, for instance, possible

choices of the parameters a and kx that allow for the existence of bound states. It was

shown that bound states do exist below the radiation continuum for all pairs (a, kx)

,i.e., no restrictions at all. When one diffraction channel is open, then bound states can

form when the pairs (a, kx) lie on the set L defined in Eq.(3–25). When two diffraction

channels are open, then the bound states are shown to only occur if the shift a is 0 or

12 and for a specific dense set of values of kx . As one goes on to higher levels of the

spectral cylinder in Fig.3-1(d), the values of kx at which bound states may exist become

more sparse and are determined by solutions of a system of diophantine equations. The

conditions under which bound states exist are formulated first for the studied system in

the continuum N ≥ 2.

The above assertions follow from the observations that, if the diffraction channels

0 ,−1 , 1 , ... are open, then at a bound state, the parameters kx , k , h , a ,R , and εc

satisfy the relations,

�(a, h, kx , k) = 0

sin(2hkz) = 0

sin(2hkz ,−1) = 0

sin(2hkz ,1) = 0

...

(3–34)

where �(a, h, kx , k) is the determinant of Eq.(3–13). The additional equations are a

result of the square integrability of the bound states on S1 ×R. Indeed, if a solution Eω to

73

Eq.(3–8) is to be square integrable on S1 × R then, the function

ξ : z 7→∫ 1

0

|Eω(x , z)|2dx

is integrable in z over R. Now,

ξ(z) → 4π2δ20(k)

∑mop

1

k2z ,m

∣∣e ihkz ,mEω(−he3) + e−i(a(kx+2πm)+hkz ,m)Eω(ae1 + he3)∣∣2, z → ∞

∑mop

1

k2z ,m

∣∣e−ihkz ,mEω(−he3) + e−i(a(kx+2πm)−hkz ,m)Eω(ae1 + he3)∣∣2, z → −∞

where the superscript ”op” in mop indicates that the summations are to be carried over all

m’s that correspond to open diffraction channels. In particular, for square integrability to

hold, each summand in the equations above must be zero. It follows that for each open

channel m0 the following system holds,e ihkz ,m0Eω(−he3) + e−i(a(kx+2πm0)+hkz ,m0

)Eω(ae1 + he3) = 0

e−ihkz ,m0Eω(−he3) + e−i(a(kx+2πm0)−hkz ,m0)Eω(ae1 + he3) = 0

(3–35)

For nontrivial solutions, the determinant of the above system must be zero. Thus,

sin(2hkz ,m0) = 0 for each open channel m0. Moreover, by considering the ratio of the field

Eω(ae1 + he3) to the field Eω(−he3) for all the open channels it is deduced from system

(3–35) that,

cos(2hkz) = cos(2hkz ,−1)e−2πia = cos(2hkz ,1)e

2πia = ... (3–36)

In particular, since for each open channel m0 we have sin(2hkz ,m0) = 0, then cos(2hkz ,m0

) =

±1. Thus e2πia = ±1, hence a = 0 or a = 12 .

The first two equations of system (3–34) determine k and h as functions of kx while

the last equations determine the values of kx . Thus as pointed out above, the values

of kx at which bound states may exist become more sparse as the number of open

diffraction channels increases.

74

In the case of two open diffraction channels, it is remarkable that these values of

kx are dense in [0,π]. A proof of this statement is given in Section 3.3.1. It is shown

there that the values of kx allowing for the existence of bound states occur in a double

sequence kn,lx where n, l are positive integers. In the leading order of R2(εc − 1), the

elements of the subsequence k2n+1,lx are shown to have the form :

k2n+1,lx ≈ π

2r 2 − 1+π5(r 2 − 1)(4r 2 − 1)4

4(2r 2 − 1)5R4(εc − 1)2, r =

l

2n + 1(3–37)

The elements of the subsequence k2n,lx are harder to derive due to a more intricate

dependence on the system parameters. For the subsequence k2n+1,lx , the corresponding

wavenumbers and distances between the arrays at which the bound states occur are

proved to be obtained by substituting k2n+1,lx into the following expressions:

k2n+1(kx) ≈ 2π + kx −8π2δ2

0(2π + kx)

2π + kx, h2n+1(kx) ≈

(2n + 1)π

2√2πkx

(1 +

πδ20(2π + kx)

kx

)(3–38)

As in the case of bound states in the continuum I, bound states in the continuum II are

shown to only occur under the following restriction on the radius and dielectric constant

of the scattering cylinders :

R√εc − 1 <

C 4√kx√

π − kx, C ≈ 2.016 (3–39)

In particular, when kx is close to 0, the quantity R√εc − 1 must be small enough in order

for bound states to exist at all. Figure 3-2(c) shows the regions in which Eq.(3–39) is

valid.

The near field amplification observed in the case of one open diffraction channel

persists when two channels are open. This can be established via an analysis similar to

that of Section 3.2.3. Figure 3-4(c) gives an example of such an amplification.

When three or more diffraction channels are open; the equations sin(2hkz ,m0) = 0 for

each open channel m0, determine the parameters kx , k and h. In fact, if n0, n1, n2, ... are

positive integers such that 2hkz = n0π, 2hkz ,−1 = n1π, 2hkz ,1 = n2π... then 2n20 = n21 + n22

75

and,

kx =n21 − n22

2n20 − n21 − n22π , h =

1

4√2

√2n20 − n21 − n22 , k =

√(n21 + n22 − 4n20)

2 + 4n21n22

2n20 − n21 − n22π

(3–40)

When only three channels are open, the integers n0, n1 and n2 are only required to

be in the order n0 > n1 ≥ n2. When four channels are open, the additional equation

sin(2hkz ,−2) = 0 in system (3–34) requires that the aforementioned integers satisfy the

system, 3n21 + n22 = 3n20 + n23

n0 ≥ n1 > n2 ≥ n3

As more diffraction channels become available, there are more and more constraints

on the integers ni , i = 0, 1, 2... Provided integers satisfying those constraints can be

found, bound states will then be formed for double arrays for which the radius R and the

dielectric constant εc satisfy �(a, h, kx , k) = 0 with h, k and kx given by Eqs.(3–40) and

a ∈ {0, 12}. From Eqs.(3–10) and (3–13), it follows that R and εc must be on curves,

2

k2R2(εc − 1)+ ln(2πR) = C(n0, n1, ...)

where k is given in Eq.(3–40) and C is some constant that depends on the integers

ni , i = 0, 1, 2, ...

3.3.1 Bound States in the Radiation Continuum II: Two Open Diffraction Chan-nels

Suppose that both the 0th and −1st diffraction channels are open ,i.e., kx ∈ (0,π]

and 2π − kx < k < 2π + kx . Conditions (3–34) then translate to the following system of

76

equations:

2 (1− cos(2πa) cos(2hkz) cos(2hkz ,−1))

kzkz ,−1

= +−

+

(1− cos(2hkz)

kz+

1− cos(2πa) cos(2hkz ,−1)

kz ,−1

)

+−

(1 + cos(2hkz)

kz+

1 + cos(2πa) cos(2hkz ,−1)

kz ,−1

)= 0

(3–41)

where ± = �∗ ∓ �c for the functions �∗ and �c of Eqs.(3–14). Also, recall that a is

necessarily 0 or 12 as derived from Eqs.(3–36).

The first equation of the system implies that +− ≥ 0. If this inequality were to be

strict, the second equation would not have held, and, therefore, +− = 0. Thus either

+ = 0 or − = 0. Note that the functions + and − cannot vanish simultaneously.

This can be verified by observing that,

+2+−2

= 2(�2

∗ +�2c

)and therefore, if the functions + and − were to vanish simultaneously; it would follow

that �c = 0. But,

�c =

∑m =0,−1

e−2hqz ,m

qz ,m> 0 if a = 0

∑m =0,−1

(−1)me−2hqz ,m

qz ,m< 0 if a = 1

2and kx = π

0 if a = 12

and kx = π

Thus (a, kx) = (12 ,π). But then kz = kz ,−1 and the first equation in system (3–41) reads,

1 + cos2(2hkz) = 0

This is impossible and therefore at a bound state + and − do not vanish simultaneously.

In particular, there are no bound states corresponding to the pair (a, kx) = (12 ,π). This is

the reason this point was removed from the third level of Fig. 3-2(a).

77

For the remainder of the discussion it is assumed that (a, kx) = (12 ,π). System

(3–41) then splits into two systems, namely,cos(2hkz) = −1

cos(2πa) cos(2hkz ,−1) = −1

+ = 0

cos(2hkz) = 1

cos(2πa) cos(2hkz ,−1) = 1

− = 0

(3–42)

Thus to each value of a corresponds a pair of systems whose solutions, if any, give rise

to bound states in the continuum. For a = 0, these systems are,

(A):

cos(2hkz) = −1

cos(2hkz ,−1) = −1

+ = 0

(B):

cos(2hkz) = 1

cos(2hkz ,−1) = 1

− = 0

(3–43a)

while for a = 12 they are,

(C):

cos(2hkz) = −1

cos(2hkz ,−1) = 1

+ = 0

(D):

cos(2hkz) = 1

cos(2hkz ,−1) = −1

− = 0

(3–43b)

The existence of solutions to the last two equations in each system is proved first. Then

the first equation of each system is added to show the existence of bound states.

For each positive integer n, the function n is defined by,

n(k , kx , a) =

+(a,

nπ

2kz ,−1

, k , kx) if n is odd and a = 0 or n is even and a =1

2

−(a,nπ

2kz ,−1

, k , kx) if n is even and a = 0 or n is odd and a =1

2(3–44a)

78

where k ∈ (2π − kx , 2π + kx). The explicit expression of n is,

n(k , kx , a) =1

2π

(1

δ0(k)+

3

2+ 2 ln(2πR)

)

+∑

m =0,−1

(1

2π(|m|+ 1)− 1− (−1)n+2a(m+1)e−nπqz ,mk

−1z ,−1

qz ,m

) (3–44b)

Then the systems formed by the last two equations of each of systems (3–43) split into

the countable set of systemsh =

nπ

2√k2 − (2π − kx)2

n(k , kx , a) = 0

n = 1, 2, 3... (3–45)

It is shown in Appendix A.3 that for each positive integer n, the function k 7→ n is

monotone decreasing on its domain (2π − kx , 2π + kx). It follows that if system (3–45)

has a solution, then this solution is unique. Moreover, such a solution will only exist if

and only if the limit of k 7→ n at 2π − kx is positive while the limit of n at 2π + kx is

negative. The first limit is,

limk→(2π−kx )+

n =1

2π

(1

δ0(2π − kx)+

3

2+ 2 ln(2πR)

)

+∑

m =0,−1

(1

2π(|m|+ 1)− 1√

(2πm + kx)2 − (2π − kx)2

) (3–46)

As in the case of bound states in the continuum I, the requirement for this limit to be

positive puts a restriction on the values of R and εc that allow for the existence of

bound states in the continuum II. An easily analyzable condition on these parameters

is obtained by following the same procedure as in Section 3.2.2. First, the positivity

79

condition is rewritten as,

∞∑m=1

(1√

(2πm + kx)2 − (2π − kx)2+

1√(2π(m + 1)− kx)2 − (2π − kx)2

− 1

π√m(m + 1)

)

+1

π

(s − 3

4− ln(2πR)

)<

2

πR2(εc − 1)(2π − kx)2

(3–47)

where

s =

∞∑m=1

(1√

m(m + 1)− 1

2

(1

m + 1+

1

m + 2

))≈ 0.691

The rearrangement of the series is made to ensure that all the summands in the series

of Eq.(3–47) are nonnegative; they vanish at kx = π. Also since the cylinders are thin,

it may be assumed that R < 12πes−

34 ≈ 0.150 so that s − 3

4− ln(2πR) > 0. Thus all

summands in Eq.(3–47) are nonnegative and hence the left hand side must be larger

than each individual summand on the right hand side. To estimate the threshold value of

R√εc − 1 below which bound states may exist, the first term (m = 1) is retained in the

sum (3–47). This term is then written in the form (π − kx)√kxg(kx) to isolate its branch

point at kx = 0 and its simple root at kx = π. The minimization of g(kx) on [0, π] produces

the estimate (3–39) where C is exactly,

C = 254π− 3

4

(min0≤t≤1

(2− t)2√3− t

(√3− t

1 +√t−

√t√

2 +√3− t

))− 12

≈ 2.016

As mentioned above, in addition to the requirement that the limit at 2π − kx of

k 7→ n be positive, one must also require that the limit at 2π + kx be negative in order

for system (3–45) to have a solution. The latter limit is,

limk→(2π+kx )−

n =

−∞ if n is odd

− nπ√2πkx

+O(1) if n is even and a = 0

− 1√3π(π − kx)

+O(1) if n is even and a = 12

80

In particular, the limit is negative if n is odd. If n is even and a = 0, the negativity

condition may be ensured by choosing kx sufficiently close to zero or by choosing n

sufficiently large. If n is even and a = 12 , then kx must be very close to π for the limit

to be negative. For parameters R and εc satisfying condition (3–47), the conditions of

existence of the solutions(kn(kx), hn(kx)

)to system (3–45) are summarized in Table 3-1.

In the leading order of δ0(2π + kx), approximate values of k2n+1 and h2n+1 are given by

Eq.(3–38)(Appendix A.4).

The approximate values for the wavenumbers k2n(kx) are more difficult to find due to

the dependence of their existence on the physical parameters of the system.

Table 3-1. Existence of solutions to systems (3–45). In particular, systems (A) and (D) in(3–43) always have solutions whereas (B) and (C) might not.

a = 0 a = 12

n odd (kn, hn) exists ∀kx ∈ (0,π] (kn, hn) exists ∀kx ∈ (0,π)n even (kn, hn) exists for n large or kx small (kn, hn) exists only for kx very close to π

Since the conditions of existence of solutions to system (3–45) are now established,

the existence of solutions to systems (3–43) can be investigated. So far only the last two

equations in each of the latter systems have been used to determine the values kn(kx)

and hn(kx) for each kx ∈ (0,π] that are susceptible to permit the existence of bound

states. It follows that the first equations in each of systems (3–43) determine the values

of kx that allow for the existence of bound states. In the coming paragraphs the set of

these values is shown to be discrete and dense in [0,π].

Let n be a positive integer for which kn(kx) exists for all kx ∈ (0,π] (kx ∈ (0,π) if

a = 12 ). Consider the function φn defined by,

φn(kx) = 2hn(kx)√k2n (kx)− k2x = nπ

√k2n (kx)− k2x

k2n (kx)− (2π − kx)2, kx ∈ (0,π)

where the value kx = π is purposely left out and will be discussed later.

Collectively, the first equations of systems (3–43) are cos (φn(kx)) = ±1 and

therefore, they have solutions if the range of φn can be shown to contain even and odd

81

integer multiples of π. That this is indeed the case follows from the continuity of φn and

its limits at 0 and π. These are,

limkx→0+

φn(kx) = ∞ and limkx→π−

φn(kx) = nπ

so that the range of φn contains the interval (nπ,∞). In particular, for each positive

integer l > n, there exists a point kn,lx ∈ (0,π) such that φn(kn,lx ) = lπ and therefore

cos(φn(k

n,lx ))= (−1)l . This establishes the existence of bound states in the continuum

II. As claimed, they exist for a discrete set of kx values in (0,π), namely, the points kn,lx .

To each of these points corresponds a specific wavenumber kn,l = kn(kn,lx ) and a specific

distance hn,l = hn(kn,lx ) at which a bound state in the continuum exists. Note that the

points kn,lx depend on R, εc and the shift a as is illustrated for instance for the points

k2n+1,lx in Eq.(3–37) (Appendix A.4). Note also that by definition, the values k2n+1,2l+1

x are

solutions to system (A) of Eqs.(3–43) while the values k2n+1,2lx are solutions to system

(B) of the same set of systems and that the two sets of points do not overlap.

As far as the case kx = π is concerned, it was established in the beginning of this

section that there can be no bound states if a = 12 . However, they do exist if a = 0.

This is because kz = kz ,−1 at kx = π so that the first equations in systems (3–43a)

are superfluous and hence the existence of solutions to systems (3–45) alone suffices

to guarantee the existence of bound states in the continuum II (Note that systems

(3–43b) become inconsistent as expected). Thus if a = 0, the list of points {kn,lx , l > n}

is to be completed by adding to it the point kx = π. Keeping with the notation, this

point is kn,nx for each n since φn can be extended to π by defining φn(π) = nπ so that

cos(φn(π)) = (−1)n. Thus the point kx = π is of infinite multiplicity in the list {kn,lx , l ≥ n}

and hence is associated with an infinite set of bound states. Indeed for each positive

integer n, sufficiently large if it is even, there exists a bound state at the wavenumber

kn,n = kn(π) and at the distance 2hn,n = 2hn(π) between the two arrays of cylinders. All

82

the other points kn,lx change with the physical parameters of the cylinders and the indices

n, l and therefore none of them is certain to be repeated.

It is remarkable that the set of points kn,lx is dense in [0, π] as is suggested by

Eq.(3–37). To demonstrate this fact, consider an arbitrary interval Iαβ = (α, β) ⊂ (0,π). It

will be shown shortly that,

limn→∞

(φ2n+1(α)− φ2n+1(β)

)= ∞ (3–48)

Therefore, however small the interval Iαβ may be, the interval φ2n+1(Iαβ) contains positive

integer multiples of π for sufficiently large n. If lπ is such a multiple, then k2n+1,lx ∈ Iαβ.

Thus the points {k2n+1,lx } are dense in (0,π].

Before establishing the limit (3–48), recall that kn = kn(kx) designates the solution

to the equation n(kn(kx), kx , a) = 0. The first task is to show that the sequence

{k2n+1(kx)}∞n=1 converges for each fixed kx . To this end, let ∞ be the function defined

by,

∞(k , kx) =1

2πδ0(k)+

∑m =0,−1

(1

2π(|m|+ 1)− 1

qz ,m

)+

1

π

(3

4+ ln(2πR)

)(3–49)

for k ∈ (2π − kx , 2π + kx). For fixed (a, kx), the function k 7→ ∞ is the pointwise limit of

the sequence of functions k 7→ n; it is also continuous and monotone decreasing on its

domain (2π − kx , 2π + kx) as is shown in Appendix A.3. As k → (2π + kx)−, ∞ → −∞

while the limit of ∞ at (2π − kx)+ is exactly limit (3–46). In particular, the latter limit is

positive as required by condition (3–47). It follows that for each kx ∈ (0,π], there exists a

unique point k∞(kx) ∈ (2π − kx , 2π + kx) such that ∞(k∞(kx), kx) = 0.

83

Now for fixed kx ,

∞(k2n+1) = ∞(k2n+1)−2n+1(k2n+1)

=

∞∑m=1

(e−(2n+1)πqz ,mk

−1z ,−1

qz ,m+e−(2n+1)πqz ,−m−1k

−1z ,−1

qz ,−m−1

)if a = 0

∞∑m=1

(e−(2n+1)πqz ,mk

−1z ,−1

qz ,m− e−(2n+1)πqz ,−m−1k

−1z ,−1

qz ,−m−1

)(−1)m+1 if a =

1

2

(3–50)

where it is understood that in the two series on the right, the terms qz ,m and kz ,−1 are

to be evaluated at k = k2n+1. The sum of the first series is obviously positive. The sum

of the second series is also nonnegative as it is the sum of an alternating series whose

terms decrease in absolute value. Thus ∞(k2n+1) ≥ 0.

As k 7→ ∞ is a decreasing function it follows that,

k2n+1(kx) ≤ k∞(kx) < 2π + kx , ∀n = 1, 2, 3... and ∀kx ∈ (0,π] (3–51)

In particular, qz ,1 =√(2π + kx)2 − k22n+1 ≥

√(2π + kx)2 − k2∞ > 0 and therefore qz ,1 does

not converge to 0 as n → ∞. Hence,

∞(k2n+1) = O

(e−(2n+1)πqz ,1k

−1z ,−1

qz ,1

)−−−→n→∞

0 = ∞(k∞)

Since the function k 7→ ∞ is continuous and bijective for each fixed kx ∈ (0,π], it

follows that k2n+1(kx) → k∞(kx) as n → ∞. Note that without condition (3–51), qz ,1

could converge to 0 causing the sequence {∞(k2n+1)}∞n=1 to diverge as indicated

by Eqs.(3–50). This is the reason the subsequence {k2n}∞n=1 had to be excluded.

For this subsequence, it can be shown through an analysis similar to the above that

k2n(kx) ≥ k∞(kx) ,∀n = 1, 2, 3... and therefore more work would be needed to show that

no subsequence of the sequence {k2n}∞n=1 converges to 2π + kx .

84

From the convergence of the sequence {k2n+1(kx)}∞n=1 it is deduced that,

limn→∞

1

(2n + 1)π

(φ2n+1(α)−φ2n+1(β)

)= φ∞(α)−φ∞(β), φ∞(kx) =

√k2∞(kx)− k2x

k2∞(kx)− (2π − kx)2

(3–52)

In Appendix A.3 the function φ∞ is shown to be strictly decreasing so that φ∞(α) −

φ∞(β) > 0. This establishes limit (3–48) and thereby the density of the points {k2n+1,lx } in

the interval (0,π]. Moreover, if |An(α, β)| is the cardinality of the set An(α, β) = {l : kn,lx ∈

(α, β)} then,

|A2n+1(α, β)| = (2n + 1)(φ∞(α)− φ∞(β)

)+ o(n)

−−−−−−−→R2(εc−1)→0

(2n + 1)√2

(√1 +

π

α−√1 +

π

β

)+ o(n)

The last limit follows from the fact that as R2(εc − 1) → 0 then k∞(kx) → 2π + kx .

Lastly, from Eqs.(3–9) and (3–12), the analytic expressions of the bound states En,l

can be obtained for positive integers n and l such that l ≥ n if a = 0 and l > n if a = 12 .

As mentioned in prior sections, each of these states can only be determined up to a

multiplicative constant which is chosen to be the value En,l(−hn,le3) of the electric field

on the cylinder at (0,−hn,l). In terms of this value, the electric field on the cylinder at

(a, hn,l) is then,

En,l(ae1 + hn,le3) = (−1)n+2a+1e iakn,lx En,l(−hn,le3)

and everywhere off the scattering cylinders it is,

En,l(r) = 2πiδ0(kn,l)En,l(−hn,le3)∑m

e ix(kn,lx +2πm)

kn,lz ,m

(e i |z+hn,l |kn,lz ,m + (−1)n+2a(m+1)+1e i |z−hn,l |kn,lz ,m

)(3–53)

where kn,lz ,m =√k2n,l − (kn,lx + 2πm)2.

85

CHAPTER 4A RESONANT GENERATION OF SECOND HARMONICS IN DOUBLE ARRAYS OF

SUBWAVELENGTH DIELECTRIC CYLINDERS

In this chapter, a nonlinear electromagnetic scattering problem is studied for two

parallel periodic arrays of dielectric cylinders with a second order nonlinear susceptibility.

For a wide range of values of the nonlinear susceptibility, the conversion rate of the

incident fundamental harmonic into the second one is shown to be as high as 40% at

the distance between the arrays as low as a half of the incident radiation wavelength. In

the framework of resonant scattering theory, the effect is attributed to the existence of

electromagnetic bound states in the radiation continuum.

4.1 The Scattering Theory

The system considered is sketched in Fig. 4-1(a). It consists of an infinite double

array of parallel, periodically positioned cylinders. The cylinders are made of a nonlinear

dielectric material with a linear dielectric constant εc > 1, and a second order

susceptibility χc ≪ 1. The coordinate system is set so that the cylinders are parallel

to the y-axis, the structure is periodic along the x-axis, and the z-axis is normal to the

structure. The unit of length is taken to be the array period, and the distance between

the two arrays relative to the period is 2h. The three coordinate axes x , y , and z , are

oriented by the unit vectors {e1, e2, e3}. In the case when the structure is illuminated

by a plane wave with electric field parallel to the cylinders (TM polarization), Maxwell’s

equations are reduced to the scalar wave equation,

1

c2∂2t

(εE +

χ

4πE 2)= �E (4–1)

where the dielectric function ε has a constant value εc > 1 on the dielectric cylinders,

and equals 1 elsewhere. Similarly, the second order susceptibility χ takes a constant

value χc ≪ 1 on the cylinders and 0 elsewhere. Due to the translation invariance of the

scattering structure in the y−direction and the TM polarization, a solution to Eq. (4–1) is

a function of x and z alone.

86

Figure 4-1. Panel (a): Double array of dielectric cylinders. The unit of length is the arrayperiod. The axis of each cylinder is parallel to the y -axis, and is at a distanceh from the x-axis.Panel (b): The scattering process for the normal incident radiation (kx = 0).The scattered fundamental harmonic is symbolized by a single headedarrow while the (generated) second harmonic radiation is symbolized by adouble headed arrow. The incident radiation wave length is such that onlyone diffraction channel is open for the fundamental harmonic while threediffraction channels are open for the second harmonic. The flux measuredthrough the faces L±1/2 : x = ±1/2 cancels out due to the Bloch periodicitycondition as explained in Appendix B.2.Panel (c): The solid and dashed curves show the position (frequencyωr = ckr ) of scattering resonances as functions of the distance between thearrays, k = kr(h). The dots on the curves indicate positions of bound statesin the radiation continuum (i.e., the values of h at which a resonance turnsinto a bound state). The solid curve connects bound states symmetricrelative to the reflection z → −z . The dashed line connects the skewsymmetric bound states. The curves are realized for R = 0.08, εc = 2, andkx = 0 (normal incidence).

It is customary to assume that the solution E is analytic in χc so that higher

harmonic effects may be studied through a power series expansion,

E = 2Re{E1e

−iωt + χcE2e−2iωt + χ2

c

(E3,1e

−iωt + E3,3e−3iωt

)+ ...

}(4–2)

where E1 is the amplitude of the fundamental harmonics in the zero order of χc , E2 is

the amplitude of the second harmonics in the first order of χc , and so on. Such a series

is obtained by perturbation theory in χc which entails first solving Eq.(4–1) in the linear

case (χc = 0) to produce a solution EL. The general solution E to the nonlinear wave

equation is then sought in the form E = EL + ENL, where ENL is the correction due to

87

nonlinear effects. If G is the Green’s function of the operator εc2∂2t − � with appropriate

(scattering) boundary conditions, then,

ENL = − χc4πc2

G[η∂2t (EL + ENL)

2]

where the second order susceptibility χ has been written in the form χ = χcη to isolate

the perturbation parameter χc . The function η is simply the indicator function of the

region occupied by the infinite double array, i.e., its value is 1 on any of the cylinders,

and 0 elsewhere. A power series expansion for the correction ENL and, hence, for the full

solution E is then obtained by the method of successive approximations.

According to scattering theory [6, 10], the kernel of the Green’s function G will be

meromorphic in k2 = ω2

c2. If the kernel has a real pole k2 = k2b > 0, it is not summable,

and therefore the successive approximations produce a diverging series, thus indicating

a non-analytic behavior of the solution in χc . As is clarified shortly, real poles correspond

to bound states in the radiation continuum so that the goal is to investigate the nonlinear

wave equation (4–1) in a small neighborhood of a real pole of the Green’s function G ,

which amounts to finding a non-analytical solution in χc . This is a crucial difference

between conventional treatments of optically nonlinear effects and the present study

from both the physical and mathematical points of view. The conclusions summarized

in the introduction stem directly from the non-analyticity of the solution of the non-linear

scattering problem in a spectral region that contains bound states in the radiation

continuum.

In general, the problem is posed as a scattering problem. An incident radiation,

Ein(r, t) = 2 cos(k · r − ωt), k = kxe1 + kze3, ck = ω,

is scattered by the double array of dielectric cylinders. The general solution to Eq. (4–1)

should then be of the form,

E(r, t) =

∞∑l=−∞

El(r)e−ilωt (4–3)

88

where E0 ≡ 0, and for all l , E−l is the complex conjugate of El (as E is real). Therefore it

is sufficient to determine only El , l ≥ 1. The amplitudes El satisfy the Bloch’s periodicity

condition derived by requiring that the full solution E to the wave equation satisfies the

same periodicity condition as the incident wave Ein, namely,

Ein(r + e1, t) = Ein

(r, t − kx

ω

)It then follows that,

El(r + e1) = e ilkxEl(r) (4–4)

This is the Bloch’s condition for the amplitude El . By Eq.(4–1), the amplitudes of different

harmonics satisfy the equations,

�El + l2k2εEl = −νl2k2(ε− 1)∑p

EpEl−p, ν =χc

4π(εc − 1)

For ease of notation, the parameter ν is often used in lieu of χc . Whenever this is the

case, it should be kept in mind that ν is proportional to χc , and is therefore very small.

The scattering theory requires that for l = ±1, the partial waves Ele−ilωt be outgoing

in the spatial infinity (|z | → ∞). The fundamental waves E±1e∓iωt are a superposition of

an incident plane wave e±i(k·r−ωt) and a scattered wave which is outgoing at the spatial

infinity. In all, the above boundary conditions lead to a system of Lippmann-Schwinger

integral equations for the amplitudes El : E1 = H(k2)[E1 + ν∑

p EpE1−p] + e ik·r

El = H((lk)2)[El + ν∑

p EpEl−p], l ≥ 2(4–5)

and E−l = E l for l ≤ −1, where H(q2) is the integral operator defined by the relation

H(q2)[ψ](r) =q2

4π

∫(ε(r0)− 1)Gq(r|r0)ψ(r0)dr0 (4–6)

in which Gq(r|r0) is the Green’s function of the Poisson operator, (q2 + �)Gq(r|r0) =

−4πδ(r − r0), with the outgoing wave boundary conditions. For two spatial dimensions,

89

as in the case considered here r = (x , z) and r0 = (x0, z0), the Green’s function is known

[13] to be Gq(r|r0) = iπH0(q|r − r0|) where H0 is the zero order Hankel function of the first

kind.

When ν = 0, the amplitudes of all higher harmonics (l ≥ 2) vanish. Therefore it

is natural to assume that |E1| ≫ |E2| ≫ |E3| ≫ · · · for a small ν. Note that this does

not generally imply that the solution, as a function of ν, is analytic at ν = 0. Under this

assumption, the solution to the system (4–5) can be approximated by keeping only the

leading terms in each of the series involved. In particular, the first equation in (4–5) is

reduced to

E1 ≈ e ik·r + H(k2)[E1] + 2νH(k2)[E 1E2

](4–7)

while the second equation becomes

E2 ≈ H((2k)2)[E2] + νH((2k)2)[E 21

](4–8)

It then follows that a first order approximation to the solution of the nonlinear wave

equation (4–1) may be found by solving the system formed by the equations (4–7) and

(4–8). To facilitate the subsequent analysis, the system is rewritten as[1− H(k2)][E1] = e ik·r + 2νH(k2)

[E 1E2

][1− H((2k)2)][E2] = νH((2k)2)

[E 21

] (4–9)

The solution of the first of Eqs.(4–9) involves inverting the operator 1 − H(k2), and

therefore necessitates a study of the poles of the resolvent [1− H(k2)]−1. Such poles are

generalized eigenvalues to the generalized eigenvalue problem,

H(k2)[E ] = E (4–10)

for fixed kx . The corresponding eigenfunctions E = Es are referred to as Siegert

states. In contrast to Siegert states in quantum scattering theory [6], electromagnetic

Siegert states satisfy the generalized eigenvalue problem (4–10) in which the operator

90

is a nonlinear function of the spectral parameter k2. Their properties were studied in

Chapter 2. In general, for fixed h, a pole to the resolvent [1− H(k2)]−1 will have the form

k2r (h) − i�(h). Scattered modes satisfy the condition k > kx (the radiation continuum).

Hence, if kr(h) > kx , then, according to scattering theory, such a pole is a resonance

pole. In the case of the linear wave equation, i.e., χc = ν = 0, the scattered flux peaks

at k = kr(h) indicating the resonance position, whereas the imaginary part of the pole

defines the corresponding resonance width (or a spectral width of the scattered flux

peak; a small � corresponds to a narrow peak). It appears that there is a discrete set

of values of the distance h = hb at which the width of some of resonances vanishes,

�(hb) = 0, that is, the resolvent [1− H(k2)]−1 has a real pole at kb = kr(hb) > kx (Chapter

3). The corresponding eigenfunctions (Siegert states) satisfying Eq.(4–10) are known

as bound states in the radiation continuum. It is the presence of such states that makes

the scattering problem defined by the system (4–9) impossible to analyze for real k

near kr(hb) by expanding the solution into a power series in χc . It should be noted that

there exists another class of bound states in this system for which k2 < k2x (Chapter 3).

These are bound states below the radiation continuum. Such states are not relevant for

the present study, and henceforth, bound states are understood as bound states in the

radiation continuum.

The objective of what follows is to show that if the parameters (h, k) of the system

are chosen appropriately in the vicinity of a critical pair (hb, kb) at which a bound state in

the radiation continuum forms, a significant portion of the energy flux of the fundamental

harmonics is transferred to second harmonics.

Note that as far as the second of Eqs.(4–9) is concerned, it can generally be

ensured that if k2b is an eigenvalue of H for a bound state, then (2kb)2 is not an

eigenvalue. Consequently, the operator (1 − H((2k)2)−1 is regular in a neighborhood

of k2b . On the other hand, the eigenvalues k2b are isolated. Therefore for k close to but

not equal to kb, the operator 1 − H(k2) is invertible. It then follows that E1 satisfies the

91

nonlinear integral equation,

E1 =(1− H(k2)

)−1[e ik·r + 2ν2H(k2)

[E 1

(1− H((2k)2)

)−1 [H((2k)2)

[E 21

]]]](4–11)

where, in accord with the notation introduced in (4–6), the function on which an operator

acts is placed in the square brackets following the operator. The operator ν2(1−H(k2))−1

that determines the “nonlinear” part of Eq. (4–11) cannot be viewed as “small” when

k2 → k2b no matter how small ν2 ∼ χ2c is, which precludes the use of a power series

representation of the solution in ν2. The behavior of the solution of the integral equation

(4–11) will be analyzed for k2 near k2b in the limit of subwavelength dielectric cylinders.

In this limit, the action of the operator H will be proved to be determined by the action

of a 2 × 2 matrix so that Eq.(4–11) becomes a system of two quadratic equations. The

analysis of this system will be considerably simplified by establishing first a particular

property of the field ratio η defined as

η(x , z) =E1(x ,−z)E1(x , +z)

(4–12)

The said property is a result of the fact that bound states, and more generally

Siegert states, have a specific parity relative to the reflection (x , z) → (x ,−z), i.e.,

they are either odd or even functions of z . This is an immediate consequence of the

commutativity of the operator H and the parity operator P defined by P[E ](x , z) =

E(x ,−z). Now, near a pole k2r (h) − i�(h), it follows from the meromorphic expansion of

[1− H(k2)]−1 that,

E1 =iC(h)

k2 − k2r (h) + i�(h)Es +O(1) (4–13)

where C(h) is some constant depending on h, and Es is an appropriately normalized

Siegert state ( Refer to Chapter 2). Consider the curve of resonances C : k = kr(h) in

the (h, k)-plane. Along C ,

E1(x , z) =C(h)

�(h)Es(x , z) +O(1) (4–14)

92

Now, as h → hb, the width �(h) goes to 0, and the Siegert state Es becomes a bound

state Eb in the radiation continuum. It then follows from Eq.(4–14) that if C(h) does not

go to zero faster than �(h) as h → hb, i.e., the pole still gives the leading contribution to

E1 in this limit, then

η(x , z) → Eb(x ,−z)Eb(x , +z)

= ±1

depending on whether the bound state Eb is even or odd in z . In the case of the linear

wave equation (ν = 0), the constant C(h) is shown to be proportional to√�(h).

Therefore, for a small ν, the assumption that C(h) does not go to zero faster than �(h)

as h → hb is justified.

The following approach is adopted to solve Eq.(4–11) near a bound state. First,

the curve of resonances C in the (h, k)-plane is found. Next, the equation is solved in

terms of the ratio η with the pair (k , h) being on the curve C . The principal part of the

amplitude E1 relative to �h = h − hb is then evaluated near a critical point (hb, kb) on

C by taking η to its limit value. This approach reveals a non-analytic dependence of the

amplitude E1 on the small parameters �h and χc of the system, which is crucial for the

subsequent analysis of the second harmonic generation.

4.2 Subwavelength Cylinders Approximation

Following the work of Chapter 3, the action of the integral operator H(q2) in

Eq. (4–11) is approximated in the limit of subwavelength dielectric cylinders. The

approximation is defined by a small parameter

δ0(q) =(qR)2

4(εc − 1) ≪ 1 (4–15)

which is the scattering phase of a plane wave with the wavenumber q on a single

cylinder of radius R. For sufficiently small R, this approximation is justified. In this

approximation the integral kernel of H(q2) is defined by (4–6) and has support on the

region occupied by cylinders. The condition (4–15) implies that the wavelength is much

larger than the radius R, and therefore field variations within each cylinder may be

93

neglected, so that ψ(x , z) ≈ ψ(n,±h) where (n,±h) are the positions of the axes of the

cylinders (n is an integer). The integration in H(q2)[ψ] yields then an infinite sum over

positions of the cylinders. By Bloch’s condition, ψ(n,±h) = e inqxψ(0,±h), so that the

function H(q2)[ψ](x , z) is fully determined by the two values ψ(0,±h). In particular,

H(q2)[ψ](0,±h) ≈ αψ(0,±h) + βψ(0,∓h) (4–16)

where the coefficients α and β are shown to be

α(q, qx) = 2πiδ0(q)

(∞∑

m=−∞

(1

qz ,m− 1

2πi(|m|+ 1)

)+

i

πln(2πR)

)

β(q, qx , h) = 2πiδ0(q)∞∑

m=−∞

e2ihqz ,m

qz ,m

where qz ,m =√q2 − (qx + 2πm)2 with the convention that if q2 < (qx + 2πm)2, then

qz ,m = i√(qx + 2πm)2 − q2. To obtain the energy flux scattered by the structure, the

action of the operator H(q2) on ψ must be determined in the asymptotic region |z | → ∞.

It is found that for |z | > h + R,

H(q2)[ψ](x , z) ≈ 2πiδ0(q)∞∑

m=−∞

(ψ(0, h)e i |z−h|qz ,m + ψ(0,−h)e i |z+h|qz ,m

) e ix(qx+2πm)

qz ,m(4–18)

4.3 Amplitudes of the Fundamental and Second Harmonics

Now that the action of the operator H(q2) has been established in (4–16) and

(4–18), the amplitudes E1 and E2 of the fundamental and second harmonics can be

determined by solving the system (4–9). As noted earlier, this will be done along a

curve C in the h, k-plane defined by k = kr(h) where kr(h) is the real part of a pole of

[1− H(k2)]−1, or equivalently, when the incident radiation has the resonant wave number

k = kr(h). This curve is obtained by studying the singularities in k2 of the operator

1 − H(k2). With this purpose, Eq. (4–10) that defines Siegert states, is written in the

subwavelength approximation according to the rule (4–16), i.e., the action of H(k2) is

94

taken at the points (0,±h). This produces the system,

[1− H ]

Eb+

Eb−

=

0

0

, H =

α β

β α

(4–19)

where Eb± = Eb(0,±h) and the functions α = α(k , kx) and β = β(k , kx) have been

defined in the previous section. In particular, bound states occur at the points (hb, kb) at

which the determinant det(1− H ) vanishes,

det

1− α −β

−β 1− α

= (1− α− β)(1− α+ β) = 0

It follows from Eq.(4–19) that the bound states for which 1 − α − β = 0 are even in z

because Eb+ = Eb− in this case. Similarly, the bound states for which 1 − α + β = 0

are odd in z . More generally, the poles of the resolvent [1 − H(k2)]−1 are complex

zeros of det(1 − H ). They are found by the conventional scattering theory formalism.

Specifically, the resonant wave numbers k2 = k2r (h) are obtained by solving the equation

Re{1− α± β} = 0 for the spectral parameter k2. According to the convention adopted in

the representation (4–13), the corresponding resonance widths are defined by

�(h) = − Im{1− α± β}∂k2Re{1− α± β}

∣∣∣∣k2=k2r (h)

where ∂k2 denotes the derivative with respect to k2. This definition of the width

corresponds to the linearization of Re{1 − α ± β} near k2 = k2r (h) as a function of

k2 in the pole factor [1 − α ± β]−1. The curves of resonances k = kr(h) > kx come in

pairs. There is a curve connecting the symmetric bound states in the h, k-plane, and

another curve that connects the odd ones.

In what follows, only the curve connecting symmetric bound states will be

considered. The other curve can be treated similarly. Panel (c) of Fig. 4-1 shows

that the first symmetric bound state occurs when the distance 2h is about half the

array period, while the skew-symmetric bound states emerge only at larger distances.

95

This feature is explained in detail in Chapter 3. So, the solution obtained near the first

symmetric bound state corresponds to the smallest possible transverse dimension of the

system (roughly a half of the wave length of the incident radiation) at which a significant

enhancement of the second harmonic generation can be observed. Thus, from now on

the curve of resonances C refers to the curve in the h, k-plane defined by the equation

Re{1− α− β} = 0. To simplify the technicalities, it will be further assumed that only one

diffraction channel is open for the fundamental harmonics, i.e., kx < k < 2π − kx . Note

that bound states in the radiation continuum exist even if more diffraction channels are

open, and the solution of the nonlinear scattering problem can also be obtained in their

vicinity by the approach developed in this study.

Let k = kr(h) be the solution of Re{1 − α − β} = 0. By making use of the explicit

form of the functions α and β for one open diffraction channel, one infers that along the

curve k = kr(h),

1− α− β = i Im{1− α− β} = −i 4πδ0(kb)kz

φ2, φ = cos(hkz), kz =√k2b − k2x

Bound states in the radiation continuum occur when the distance h satisfies the equation

φ = 0, i.e., h√k2r (h)− k2x = (n − 1/2)π with n being a positive integer. Its solutions

h = hb(n) define the corresponding values of the wave numbers of the bound states,

kb(n) = kr(hb(n)). So, the sequence of pairs {(hb(n), kb(n))}∞n=1 indicates positions of

the bound states on the curve C . In the limit h → hb(n) along C , the function φ has the

asymptotic behavior,

φ = (−1)nkz ,b�h + o(�h), kz ,b = kz |h=hb(n), �h = h − hb

The objective is to determine the dependence of the amplitudes E1 and E2 on the

parameters �h and χc which are both small, and, in particular, to investigate the

outgoing flux of the second harmonics as a function of �h and χc .

96

To this end, let E1± = E1(0,±h) be the values of the field E1 on the axes of

the cylinders at (0,±h), and E2± = E2(0,±h) be the values of the field E2 on the

same cylinders. In the subwavelength approximation, these values fully determine the

scattered field as is shown later and, hence, have to be found first. Applying the rule

(4–16) to evaluate the action of the operator H(q2) in the system (4–9), the first equation

of the latter becomes,

[1− H ]

E1+

E1−

= 2νH

E 1+E2+

E 1−E2−

+

e+ihkz

e−ihkz

(4–20a)

Similarly, the second of Eqs.(4–9) yields,

[1− H2]

E2+

E2−

= νH2

E 21+

E 21−

, H2 = H (2k , 2kx) (4–20b)

As stated above, the resolvent [1 − H((2k)2)]−1 is regular in a neighborhood of kb so

that Eq. (4–20b) can be solved for E2±, which defines the latter as functions of E1±. The

substitution of this solution into Eq.(4–20a) gives a system of two nonlinear equations

for the fields E1±. Adding these equations and replacing the field E1− by its expression

E1− = η(0, h)E1+ in terms of the field ratio of Eq.(4–12) yields the following implicit

relation between the field E1+ and its amplitude |E1+|:

E1+ = − φν2

ζ|E1+|2 + φ2ξ

(4–21)

where φ = cos(hkz) and ν = χc

4π(εc−1)are small and, in terms of the field ratio η ≡ η(0, h),

the values of ζ and ξ read,

ξ = i2πδ0(k)

kz(1 + η),

1

ζ=

(1 + i

4πδ0(k)

kzφ2

)(a + bη2 + η

(b + aη2

))(4–22)

97

with a and b being defined by the relation,

[1− H2]−1H2 =

a b

b a

(4–23)

In particular, ζ and ξ are continuous functions of η and φ. In Appendix B.1 it is shown

that if ζb and ξb are the respective limits of ζ and ξ as h → hb along the curve of

resonances C , then these limits are nonzero. It follows then that Eq.(4–21) for E1+ is

singular in both ν and φ when these parameters are small, i.e., in the limit (ν,φ) →

(0, 0). Furthermore, there is no way to solve the said equation perturbatively in either

of the parameters. A full non-perturbative solution can be obtained using Cardano’s

method for solving cubic polynomials. Indeed, by taking the modulus squared of both

sides of the equation, it is found that,

X 3 + 2φ2

ν2Re{ζξ}X 2 +

φ4

ν4|ζξ|2X − φ2

ν4|ζ|2 = 0, X = |E1+|2 (4–24)

The solution to this cubic equation is obtained in Appendix B.3. It is proved there

that Eq. (4–24) admits a unique real solution so that there is no ambiguity on the choice

of E1+. In the vicinity of a point (hb, kb) along the resonance curve C , the field E1+ is

found to behave as,

|E1+| =|�h|1/3

χcτ(�h,χc) (4–25)

Recall that �h = h − hb. An explicit form of the function τ(�h,χc) is given in Appendix

B.3 (Eq. (B–8)). It involves combinations of the square and cube roots of functions in �h

and χc and has the property that τ(�h,χc) → 0 as (�h,χc) → (0, 0) (in the sense of the

two-dimensional limit). In the limit h → hb, the field ratio η approaches 1 for a symmetric

bound state as argued earlier. Therefore it follows from Eqs.(4–20b) that E2± ∼ νE 21+

because the matrix (4–23) exists at h = hb. Since ν ∼ χc , relation (4–25) leads to the

conclusion thatE2±

E1+

= O(|�h|1/3)

98

Thus, the approximation |E1| ≫ |E2| used to truncate the system (4–9) remains

valid for h close to the critical value hb despite the non-analyticity of the amplitudes at

(�h,χc) = (0, 0).

4.4 Flux Analysis: The Conversion Efficiency

For the nonlinear system considered, even though Poynting’s theorem takes a

slightly different form as compared to linear Maxwell’s equations, the flux conservation

for the time averaged Poynting vector holds. The scattered energy flux carried across

a closed surface by each of the different harmonics adds up to the incident flux across

that surface. The flux conservation theorem is stated in Appendix B.2. Consider a

closed surface that consists of four faces, L± = {(x , z)| − 12≤ x ≤ 1

2, z → ±∞}

and L±1/2 = {(x , z)|x = ±1/2} as depicted in Fig. 4-1(b). As argued in Appendix B.2,

the scattered flux of each l th-harmonics across the union of the faces L±1/2 vanishes

because of the Bloch condition (and so does the incident flux for any kx ). Therefore

only the flux conservation across the union of the faces L± has to be analyzed. If σl

designates the ratio of the scattered flux carried by the l th-harmonics across the faces

L± to the incident flux across the same faces, then∑

l≥1 σl = 1. Thus, for l ≥ 2, σl

defines the conversion ratio of fundamental harmonics into the l th-harmonics.

In the perturbation theory used here, only the ratios σ1 and σ2 may be evaluated.

By laborious calculations it can be shown that σ1 + σ2 ≤ 1 as one would expect

(Appendix B.2). Hence, the efficiency of converting the fundamental harmonic into the

second harmonic is simply determined by the maximum value of σ2 as a function of the

parameter h at a given value of the nonlinear susceptibility χc .

The ratio σ2 is defined in terms of the scattering amplitudes of the second harmonic,

i.e., by the amplitude of E2 in the asymptotic region |z | → ∞:

E2(r) →

∑mop,sh

Rshm e

ir·k−m,sh , z → −∞

∑mop,sh

T shm e ir·k

+m,sh , z → +∞

(4–26)

99

where k±m,sh = (2kx + 2πm)e1 ± k shz ,me3 is the wave vector of the second harmonic in

the mth open diffraction channel. Recall that the mth channel is open provided (2k)2 >

(2kx + 2πm)2 and in this case k shz ,m =√(2k)2 − (2kx + 2πm)2, while if the channel is

closed, then k shz ,m = i√(2kx + 2πm)2 − (2k)2. In the asymptotic region |z | → ∞, the field

in closed channels decays exponentially and, hence, the energy flux can only be carried

in open channels to the spatial infinity. The summation in Eqs.(4–26) is taken only over

those values of m for which the corresponding diffraction channel is open for the second

harmonic, which is indicated by the superscript “op, sh” in the summation index mop,sh.

Note that there is more than one open diffraction channel for the second harmonic even

though only one diffraction channel is open for the fundamental one. For instance, if the

x−component of the wave vector k, i.e., kx , is less than π2, there are 3 open diffraction

channels for the second harmonic, the channels m = 0, m = −1, and m = 1. These

three directions of the wave vector of the second harmonic propagating in each of the

asymptotic regions z → ±∞ are depicted in Fig. 4-1(b) by double-arrow rays. Thus,

in terms of the scattering amplitudes introduced in Eqs. (4–26), the ratio of the second

harmonic flux across L± to the incident flux is

σ2 =1

2kz

∑mop,sh

k shz ,m(|Rsh

m |2 + |T shm |2)

The scattering amplitudes Rshm and T sh

m are inferred from Eq.(4–8) in which the rule

(4–18) is applied to calculate the action of the operator H((2k)2) in the far-field regions

|z | → ∞: Rshm =

2πiδ0(2k)

k shz ,m

[(E2+ + νE 2

1+)eihkshz ,m + (E2− + νE 2

1−)e−ihkshz ,m

]T shm =

2πiδ0(2k)

k shz ,m

[(E2+ + νE 2

1+)e−ihkshz ,m + (E2− + νE 2

1−)eihkshz ,m

]Since η → 1 as h → hb along C , the principal part of σ2 in a vicinity of a bound state

along the curve C is obtained by setting E1− = E1+ in Eq. (4–20b), solving the latter for

100

E2±, and substituting the solution into the expression for σ2. The result reads

σ2 = Cbν2|E1+|4 (4–27)

where Cb is a constant obtained by taking all nonsingular factors in the expression of σ2

to their limit as h → hb, which gives

Cb =

[(16πδ0(k))

2

kz|1 + a + b|2

∑mop,sh

cos2(hk shz ,m)

k shz ,m

](h,k)=(hb,kb)

for a and b defined in Eq.(4–23). Using the identity |E1+|4 = |E1+|2|E1+|2, and substituting

Eq.(4–21) into one of the factors |E1+|2, the conversion ratio σ2 is expressed as a

function of a single real variable,

σ2(u) = C ′b

u

|u + ζbξb|2, u =

(ν|E1+|φ

)2

(4–28)

where C ′b = Cb|ζb|2 is a constant, and ξ and ζ in Eq.(4–22) have been taken at their

limits as h → hb to obtain the principal part of σ2. The function u 7→ σ2(u) on [0,∞)

is found to attain its absolute maximum at u = |ξbζb|. This condition determines the

distance 2h between the arrays at which the conversion rate is maximal for given

parameters R, εc and χc of the system. Indeed, since u ∼ |E1+|2 should also satisfy the

cubic equation (4–24), the substitution of u = |ζbξb| into the latter yields the condition

ν2

φ4= 2|ξb|2 (|ξbζb|+ Re{ξbζb}) (4–29a)

In particular, in the leading order in δ0(k), the optimal distance 2h between the two

arrays is given by the formula,

(h − hb)4 =

χ2c

8π5kz ,b(kbR)6(εc − 1)5(4–29b)

where as previously, kz ,b =√k2b − k2x .

The maximum value σ2,max of the conversion ratio σ2 is the sought-for conversion

efficiency. An interesting feature to note is that σ2,max = σ2(|ζbξb|) is independent

101

of the nonlinear susceptibility χ2 because the constants C ′b, ξb, and ζb are fully

determined by the position of the bound state (kb, hb). In other words, if the distance

between the arrays is chosen to satisfy the condition (4–29a), the conversion efficiency

σ2,max is the same for a wide range of values of the nonlinear susceptibility χ2. This

conclusion follows from two assumptions made in the analysis. First, the subwavelength

approximation should be valid for both the fundamental and second harmonics, i.e.,

the radius of cylinders should be small enough. Second, the values of h − hb and ν (or

χc ) must be such that the analysis of the existence and uniqueness of |E1+| given in

Appendix B.3 holds, that is, Eq. (4–24) should have a unique real solution under the

condition (4–29a). The geometrical and physical parameters of the studied system can

always be chosen to justify these two assumptions as illustrated in Fig. 4-1.

Panels (a) and (b) of Fig. 4-2 show the conversion efficiency σ2,max for the first three

symmetric bound states n = 1, 2, 3, as, respectively, a function of the cylinder radius R

when kx = 0 and of kx when R = 0.15. For all curves presented in the panels, εc = 1.5.

The values of σ2,max are evaluated numerically by Eq. (4–28) where u = |ξbζb|. The

solid parts of the curves in Panel (a) correspond to the scattering phase δ0(k) < 0.25

with k = kb. Note that the wavelength at which the second harmonic generation is most

efficient is the resonant wavelength defined by k = kr(h) where h satisfies the condition

(4–29a). For a small χc , the scattering phase at the resonant wavelength can well be

approximated as δ0(k) ≈ δ0(kb). The condition δ0(kb) < 0.25 ensures that the scattering

phase for the second harmonic satisfies the inequality δ0(2kb) = 4δ0(kb) < 1 otherwise

the validity of the subwavelength approximation cannot be justified. The dashed parts

of the curves in Panels (a) of Fig. 4-2 correspond to the region where δ0(kb) > 0.25.

Panels (a) and (b) of the figure show that the conversion efficiency can be as high as

40% for a wide range of the incident angles and values of the cylinder radius. Such a

conversion efficiency is comparable with that achieved in optically nonlinear crystals

at a typical beam propagation length (active length) of a few centimeters, whereas

102

Figure 4-2. Panel (a): The conversion efficiency is plotted against the cylinder radius Rfor the critical points (hb(n), kb(n)), n = 1, 2, 3, and εc = 1.5 and kx = 0 (thenormal incidence). The dashed parts of the curves indicate the regionswhere δ0(k) > 0.25 and, hence, δ0(2k) > 1, i.e., the subwavelengthapproximation becomes inapplicable for the second harmonic.Panel (b): The conversion efficiency is plotted against kx for the criticalpoints (hb(n), kb(n)), n = 1, 2, 3. The curves are realized for R = 0.15 andεc = 1.5.Panel (c): The region of validity of the developed theory for the first boundstate hb(1) ≈ 0.259. The shadowed part of the (ν, h)−plane is defined by thecondition τ+ + τ− > 0 under which, according to Eq.(B–8), the amplitude|E1+| exists and unique as explained in Appendix B.3. The plot is realized forR = 0.1, εc = 2 and kx = 0. The parabola-like curve is an actual boundary ofthe shadowed region; the top horizontal line represents no restriction. Sothere is a wide range of the physical and geometrical parameters within theshadowed region which satisfy (4–29a). The regions of validity for otherbound states looks similar.

here the transverse dimension 2h of the system studied here can be as low as a half

of the wavelength, i.e., for an infrared incident radiation, 2h is about a few hundred

nanometers. Indeed, as one can see in Fig. 4-2(c), the first bound state occurs at

hb(1) ≈ 0.259 and kb ≈ 2π which corresponds to the wavelength λb = 2π/kb ≈ 1.

The stated conversion efficiency can be fairly well estimated in the leading order of

δ0(k):

σ2,max = σ2(|ζbξb|) ≈ 8πδ0(k)∑mop,sh

cos2(hk shz ,m)

k shz ,m

∣∣∣∣∣(h,k)=(hb,kb)

(4–30)

103

Suppose that only one diffraction channel is open for the incident radiation. Then

m = 0,±1 in Eq.(4–30) (three open channels for the second harmonics). Let σ02,max

denote the term m = 0, i.e. σ02,max is the second harmonic flux in which the contribution

of the channels with m = ±1 is omitted. In particular, σ02,max < σ2,max. One infers from Eq.

(4–30) that,

σ02,max ≈4πδ0(kb)

kz ,bcos2(2hbkz ,b)

As the pair (hb, kb) at which a symmetric bound state is formed satisfies the equation

cos(hbkz ,b) = 0, it follows that cos(2hbkz ,b) = −1. Hence,

σ02,max ≈k2bkz ,b

πR2(εc − 1)

The wavenumbers kb at which the bound states occur lie just below the diffraction

threshold 2π − kx , i.e., kb / 2π − kx (Chapter 3). So that in the case of normal incidence

(kx = 0), the above estimate becomes,

σ02,max ≈ 2π(πR2)(εc − 1)

with δ0(2π) ≪ 1. If, for instance, R = 0.15 and εc = 2, then,

σ02,max ≈ 44%

for δ0(2π) ≈ 0.22.

It is noteworthy to emphasize the following features of the proposed mechanism

to generate higher harmonics that are to be contrasted with the conventional methods.

First, the necessity to fulfill the phase matching condition, much needed in optically

nonlinear crystals, has been eliminated. The reason is that the second harmonic is

generated in regions (cylinders) of dimensions much smaller than the wave length and,

hence, the phase mismatch in propagation of the first and second harmonics due to

the difference in the corresponding refraction indices does not even occur. Second, the

process does not require any focusing of the incident beam. Instead, if the geometrical

104

parameters of the system are set to maximize the conversion rate, the focusing occurs

within the structure automatically, and the maximal conversion rate of about 40% is

achieved, even though χcEi ≪ 1 for the incident radiation. Third, the maximum value

of the conversion rate depends weakly on the nonlinear susceptibility. This provides a

possibility for the frequency conversion in lower power light beams. Fourth, an active

length at which the conversion rate is maximal is close to 2hb whose smallest value

for the system studied is roughly a half of the wave length of the incident light, while

an effective conversion in a slab of an optical nonlinear material requires a length

varying between a few millimeters to a few centimeters. This means that the conversion

can effectively be done at nanoscales for a visible light. The very existence of all the

aforementioned features is essentially attributed to the existence of resonances with

the vanishing width or bound states in the radiation continuum for the scattering system

studied, and would not be possible otherwise.

105

APPENDIX ACOMPLEMENTS I

A.1 The Lippmann-Schwinger Integral Equation

In this section of the appendix, it is proved that the solution to the Lippmann-Schwinger

integral equation in Eq.(2–5) solves Eq.(2–2). It should be stressed that this equation

is to be understood in the distributional sense. This is because the potential under

consideration is neither compactly supported nor does it vanish at infinity so that the

usual methods that establish this equation cannot be used [10]. Thus, the proof will be

done by considering the functions involved as distributions acting on smooth functions of

compact support.

To do so, only locally integrable solutions to Eq.(2–2) are sought. In this setting, all

the functions involved in Eq.(2–5), namely; e ik·r, Eω, εEω, (ε− 1)Eω, Gk and ((ε− 1)Eω) ∗

Gk , are locally integrable.

Let then φ be a smooth function of compact support and Eω be a locally integrable

solution to Eq.(2–5). Then

⟨�Eω + k2εEω,φ⟩ = k2⟨εEω,φ⟩ − k2⟨Ei ,φ⟩+k2

4π⟨((ε− 1)Eω) ∗ Gk , �φ⟩ (A–1)

where Ei(r) = e ik·r. Since none of the distributions (ε − 1)Eω and G is compactly

supported, the convolution used here is to be understood in the sense of the usual

convolution of functions. Therefore, the last integral of Eq.(A–1) may be interpreted as

⟨((ε− 1)Eω) ∗ Gk , �φ⟩ =∫(ε(r0)− 1)Eω(r0)⟨Gr0, �φ⟩dr0 (A–2)

where Gr0(r) = Gk(r|r0) and satisfies the distributional Helmholtz equation

�Gr0 + k2Gr0 = −4πδr0 (A–3)

Therefore

⟨Gr0, �φ⟩ = ⟨�Gr0,φ⟩ = −k2⟨Gr0,φ⟩ − 4π⟨δr0,φ⟩

106

It follows that

⟨((ε− 1)Eω) ∗ Gk , �φ⟩ = −k2⟨((ε− 1)Eω) ∗ Gk ,φ⟩ − 4π⟨(ε− 1)Eω,φ⟩ (A–4)

Thus ⟨�Eω + k2εEω,φ⟩ = 0 for all test functions φ.

Note that the only difference between the proof that Eq.(2–5) solves Eq.(2–2) in the

case of the finite array and that of an infinite array is the way Eq.(A–4) is derived from

Eq.(A–2). Indeed, if the array of cylinders is finite then the convolution of Eq.(A–2) is

in the distributional sense as (ε − 1)Eω would be a compactly supported distribution.

Therefore one could establish Eq.(A–4) immediately from Eq.(A–3) and the identity of

distributional convolution

⟨((ε− 1)Eω) ∗ Gk , �φ⟩ = ⟨((ε− 1)Eω) ∗ �Gk ,φ⟩ (A–5)

This identity is not immediate in the case of the infinite array because, as mentioned

above, none of the convoluted functions has compact support. In fact, the function

((ε− 1)Eω) ∗ Gk is a conditionally convergent series so that integration by parts cannot

be applied to establish Eq.(A–5).

A.2 Solution of the Lippmann-Schwinger Integral Equation in the Zero RadiusApproximation

Here an approximate solution to the Lippmann-Schwinger integral equation is

established in the small radius approximation. In the usual theory of scattering from

small particles [15–18], the problem is solved with high accuracy by assuming that far

from the scattering region the solution is a linear superposition of the waves scattered

by each individual particle. Due to the cylindrical geometry of the dielectric scatterers,

it turns out that similar approximations may be made to solve the scattering problem on

the double array but with solutions which are valid everywhere off the scatterers even in

the region between the two grating structures. The validity of solution in this extended

107

region can be used to find the so-called hot spots (Fig. 3-3) where the magnitude of the

electromagnetic fields peaks.

The structure considered is shown in Fig. 3-1(a). The cylinders in the structure

are labeled as Cm,n where n is either 1 or −1 depending on whether the cylinder is on

the right or left array. The integer m refers to the x-coordinate of the cylinder’s axis. In

particular, for the right array cylinders, the axes are positioned at rm,1 = (m + a)e1 + he3

and those of the left array are at positions rm,−1 = me1 − he3.

The Lippmann-Schwinger integral equation may then be written as a sum over all

cylinders as

Eω(r) = e ik·r +ik2(εc − 1)

4

∑m,n

∫Cm,n

Eω(r0)H0(k |r − r0|)dr0 (A–6)

Far from the scatterers, each of the integrals is well approximated through the mean

value theorem by∫Cm,n

Eω(r0)G(r|r0)dr0 ≈ iπ2R2e imkxH0(k |r − rm,n|)Eω(r0,n)

so that the far field may be expressed in terms of the fields on the cylinders C0,±1 as

Eω(r) = e ik·r + iπδ0(k)

∑n=±1

Eω(r0,n)

∞∑m=−∞

e imkxH0(k |r − rm,n|) for δ0(k) =

1

4k2R2(εc − 1)

(A–7)

The claim is that this approximation remains valid in the near region too. This may be

established by means of the Bessel function expansions of the field inside the cylinders

and the Hankel function H0. To this end, let r0 be a position vector on the cylinder Cm,n,

then r0 = rm,n + u with u = |u| ≤ R. If r is a position vector off the scatterers, then

|r − rm,n| > R and therefore

H0(k |r − r0|) =∞∑

ν=−∞

e iν(θm,n−θ)Jν(ku)Hν(k |r − rm,n|)

108

where θ and θm,n are the angles between the x-axis and the vectors u and r − rm,n

respectively. On the other hand, the field inside the cylinder Cm,n is given by

Eω(r0) = e imkx

∞∑ν=−∞

αν,neiνθJν(k

′u) (A–8)

with k ′ = nck for the index of refraction nc of the cylinders and the coefficients αν,n given

by

αν,n =1

2πJν(k ′R)

∫ 2π

0

e−iνθEω(Rer + r0,n)dθ

Here er = e1 cos θ + e3 sin θ is the usual radial vector of polar coordinates. In particular,

αν,n is at most of the order of (kR)−|ν| in the limit of thin cylinders.

It follows that∫Cm,n

Eω(r0)G(r|r0)dr0 = 2π2ie imkx

∞∑ν=−∞

αν,neiνθm,nHν(k |r − rm,n|)

∫ R

0

Jν(k′u)Jν(ku)udu

As the integral in the above series is of the order of (kR)2|ν|+2, it is then justified to

approximate the series by its 0th summand in the limit kR ≪ 1, so that∫Cm,n

Eω(r0)G(r|r0)dr0 ≈ iπ2R2e imkxα0,nH0(k |r − rm,n|) (A–9)

The value of α0,n may then be recovered by setting u = 0 in Eq.(A–8). It is Eω(r0,n).

This establishes Eq.(A–7) everywhere off the scatterers. In particular, the fields are

determined by the knowledge of their values Eω(r0,±1) on the cylinders C0,±1 alone.

To determine the values Eω(r0,±1), let n be either 1 or −1 and, r0,n be substituted for r

in Eq.(A–6). The latter equation becomes,

Eω(r0,n) = e ik·r0,n +ik2(εc − 1)

4

(∫C0,n

Eω(r0)H0(k |r0,n − r0|)dr0

+∑m =0

∫Cm,n

Eω(r0)H0(k |r0,n − r0|)dr0

+∑m

∫Cm,−n

Eω(r0)H0(k |r0,n − r0|)dr0) (A–10)

109

where the integral over C0,n has been isolated due to the singularity of its integrand at

r0 = r0,n. To approximate this particular integral, Eq.(A–8) is used to obtain in the leading

order of kR;∫C0,n

Eω(r0)H0(k |r0,n − r0|)dr0 =2πEω(r0,n)

∫ R

0

J0(k′u)H0(ku)udu

≈ πR2Eω(r0,n)

(1 +

2i

π

(γ + ln

(kR

2

)− 1

2

))where γ is the Euler constant. All the other integrals in Eq.(A–10) obey the estimate

(A–9). By taking n successively equal to 1 then to −1, the following system is obtained:�0Eω(ae1 + he3) + �+Eω(−he3) =

i

2πδ0(k)

e i(akx+hkz )

�−Eω(ae1 + he3) + �0Eω(−he3) =i

2πδ0(k)

e−ihkz

(A–11)

The functions �0,�+ and �− are

�0(k , kx) =i

2πδ0(k)

+1

2

(∑m =0

e imkxH0(k |m|) + 1 +2i

π

(γ + ln

(kR

2

)− 1

2

))

�±(a, h, k , kx) =1

2

∑m

e imkxH0(k |(m ∓ a)e1 + he3|)

The variants of these functions in Eqs.(3–10) are obtained through the formulas,

1

2

∞∑m=−∞

e imkxH0(k |r −me1|) =∞∑

m=−∞

e i(x(kx+2πm)+|z |kz ,m)

kz ,m, r = 0 (A–12a)

1

2

∑m =0

e imkxH0(k |m|) =∞∑

m=−∞

(1

kz ,m− 1

2π(|m|+ 1)

)− 1

2− i

π

(γ + ln

(k

4π

)− 1

2

)(A–12b)

Relation (A–12a) can be proved by substituting the plane wave representation of the

Hankel function

H0(kr) =i

π2

∫ ∫e i(xKx+zKz )

k2 − K 2x − K 2

z + iηdKzdKx , η → 0+

into the left side of (A–12a) and carrying out the integration with respect to Kz followed

by an application of the Poisson summation formula to evaluate the integral with respect

110

to Kx . Relation (A–12b) is the obtained from (A–12a) in the limit r → 0, where for the

term m = 0 in the left side of Eq.(A–12a), the asymptotic expansion of the Hankel

function for a small argument has to be used.

When the determinant of the system (A–11) is nonzero, Eq.(3–2) has a unique

solution. Otherwise, the homogeneous Lippmann-Schwinger equation admits nonzero

solutions: the bound states. These states can then be arbitrarily superposed to obtain

the general solution. The various expressions for these bound states in Sections 3.2

and 3.3.1 are obtained by applying formula (A–12a) to Eq.(A–7) in the absence of the

incident wave, i.e., by omitting the term e ik·r.

A.3 Complements on Bound States in the Continuums I and II

This section of the Appendix gives some of the technical details omitted in

sections 3.2.2 and 3.3.1. First, Eq.(3–24) is established for the sequence {cm}∞m=1

defined in Eq.(3–22). In the case under consideration, the sequence {qz ,m} is in the

order,

qz ,−1 ≤ qz ,1 < qz ,−2 ≤ qz ,2 < ...

with equalities occurring when kx = 0. Since the function f : t ∈ (0,∞) 7→ t−1e−2ht

is strictly decreasing and cm = f (qz ,−m) − f (qz ,m), it follows that cm ≥ 0 with equality

holding only if kx = 0.

Now suppose that kx = 0 and, hence, cm > 0, m = 1, 2, 3.... To complete the proof

of Eq.(3–24), it suffices to show that,

cm+1

cm≤ e−4πh and lim

m→∞

cm+1

cm= e−4πh (A–13)

To establish the first of conditions (A–13), the ratio of cm+1 to cm is rewritten as,

cm+1

cm=α−m−1

α−m

1− αm+1

α−m−1

1− αm

α−m

, αm =e−2hqz ,m

qz ,m(A–14)

111

Next, the following chain of conclusions holds:qz ,m + qz ,−m−1 ≥ qz ,−m + qz ,m+1

qz ,mqz ,−m−1 ≥ qz ,−mqz ,m+1

⇒ α−mαm+1

α−m−1αm

≥ 1 ⇒1− αm+1

α−m−1

1− αm

α−m

≤ 1

Therefore,cm+1

cm≤ α−m−1

α−m

≤ e2h(qz ,−m−qz ,−m−1)

The first of conditions (A–13) then follows as qz ,−m − qz ,−m−1 ≤ −2π. The limit in (A–13)

follows from (A–14) and the limits,

limm→∞

α−m−1

α−m

= e−4πh limm→∞

αm

α−m

= e−4hkx = 1

Second, the formula (3–23) is proved. This is done by a repetitive application of

Abel’s partial summation formula. Let un be defined for each n = 0, 1, 2, ... by,

un =

∞∑m=1

cm+n sin(2πam)

The objective is to show that another expression of −u0 is (3–23). By Abel’s partial

summation formula,

un =

∞∑m=1

(cm+n − cm+n+1)sin(πam) sin(πa(m + 1))

sin(πa)

= cot(πa)∞∑

m=1

(cm+n − cm+n+1) sin2(πam) +

1

2

∞∑m=1

(cm+n − cm+n+1) sin(2πam)

= cot(πa)∞∑

m=1

(cm+n − cm+n+1) sin2(πam) +

1

2un −

1

2un+1

Thus,

u0 = (−1)N+1uN+1 + 2cot(πa)∞∑

m=1

(cm + 2

N∑n=1

(−1)ncm+n + (−1)N+1cm+N+1

)sin2(πam)

For all N = 1, 2, 3, ... By using the first of conditions (A–13) it is straightforward that

uN+1 → 0 as N → ∞, and Eq.(3–23) follows.

112

Third, the functions k 7→ n(k , kx , a) defined in Eq.(3–26) are proved to be

monotonically decreasing, i.e., ∂kn < 0. This derivative reads

∂n

∂k= − 1

πkδ0(k)−∑m =0

k

q3z ,m

(1− (−1)n cos(2πam)e−nπqz ,mk

−1z

(1 + nπ

(qz ,m

kz+q3z ,m

k3z

)))

Now, if t > 0 and n is a positive integer; then e−t(1 + t

(1 + (nπ)−2t2

))≤ 1. Setting t =

nπqz ,mk−1z shows that all summands are positive and hence ∂kn < 0, ∀n = 1, 2, 3, ....

The functions k 7→ n(k , kx) and k 7→ ∞(k , kx) defined in Eqs.(3–44) and (3–49) are

shown to be decreasing in a similar fashion.

Lastly, the function kx 7→ φ∞ defined in Eq.(3–52) is shown to be monotonically

decreasing on (0,π). To establish this fact, note that since ∂k∞ < 0, the implicit

function theorem implies that the function kx 7→ k∞(kx) is continuously differentiable and

k ′∞(kx) = −∂kx∞(∂k∞)−1. Now,

∂∞

∂kx=

∞∑m=1

(2πm + kx√

(2πm + kx)2 − k2− 2πm + 2π − kx√

(2πm + 2π − kx)2 − k2

)> 0

Hence k ′∞(kx) > 0. By logarithmic differentiation, it follows that,

φ′∞(kx)

φ∞(kx)=

k∞k′∞(k2x − (2π − kx)

2)− kx

(k2∞ − (2π − kx)

2)− (2π − kx)

(k2∞ − k2x

)(k2∞ − k2x ) (k

2∞ − (2π − kx)2)

< 0

since 0 < kx < 2π − kx < k∞(kx).

A.4 Approximations

In this section of the Appendix, it is outlined how the approximations in Eqs.(3–16),

(3–30), (3–38) and (3–37) can be obtained. The computations of the wavenumbers in

the first three of the latter equations being similar, so only Eqs.(3–30) is established, and

the discussion is finished by proving Eq.(3–37). Suppose first that only one diffraction

channel is open and (a, kx) is in the set L of Eq.(3–25). The objective is to approximate

the value kn ∈ (kx , 2π − kx) such that n(kn, kx , a) = 0 for the function n defined in

Eq.(3–26). This is done by identifying the leading terms in n near the wavenumber kn.

113

Since the dielectric cylinders forming the double array are assumed to be thin in

comparison to the wavelength ,i.e., kR ≪ 1, the quantity (2πδ0(k))−1 in the expression

of n is large. Consequently, the wavenumber kn such that n(kn, kx , a) = 0 must be

close to the diffraction threshold 2π − kx so that the term q−1z ,−1 in n is large enough

to compensate for the magnitude of (2πδ0(k))−1. Thus a first approximation for the

wavenumber kn may be found by solving the equation,

1

2πδ0(k)− 1− (−1)n cos(2πa)e−nπqz ,−1k

−1z

qz ,−1

= 0

which is obtained by keeping only the leading terms in the expression of n near kn.

When (−1)n cos(2πa) = 1, then the equation becomes,

1

2πδ0(k)− 1− e−nπqz ,−1k

−1z

qz ,−1

= 0 (A–15)

In particular if n is not large, this equation has no roots since as qz ,−1 becomes smaller,

then the second summand gets closer to nπk−1z and hence is much smaller than the first

summand. This was to be expected since in the case (−1)n cos(2πa) = 1, it was already

established that bound states exist only for sufficiently large n. Also, the initial integer n

at which the wavenumber kn exists for (−1)n cos(2πa) = 1 grows as kR → 0. This makes

it impossible to provide a good approximation for the exponential term in Eq.(A–15) that

would allow a perturbative solution of the said equation. This complexity disappears

when (−1)n cos(2πa) = 1. In this case and for n not too large, the approximation

e−nπqz ,−1k−1z ≈ 1 is valid and Eq.(A–15) becomes,

1

2πδ0(k)− 1− (−1)n cos(2πa)

qz ,−1

= 0

Thus,

kn ≈√(2π − kx)2 − 4π2(1− (−1)n cos(2πa))2δ2

0(2π − kx) (A–16)

114

The first of Eqs.(3–30) follows by keeping the first two terms in a series expansion of the

right-hand side of Eq.(A–16) in powers of δ0(2π − kx). The distance hn in Eqs.(3–30) can

then be derived as indicated by system (3–27).

Eqs.(3–16) and (3–38) are obtained through similar treatments of the functions

defined in Eqs.(3–15) and (3–44) respectively. In particular, for the bound states

below the continuum, it appears possible to give only the approximate value of the

wavenumber k+ while the wavenumber k− eludes the perturbation method due to

a complicated dependence of its existence condition on the size of the cylinders.

Similarly, in the case of two open channels, it turns out to be only possible to solve for

the wavenumbers k2n+1 whose existence is not subject to changes in cylinder sizes.

Expressions analogous to Eq.(A–16) for the wavenumbers k+ and k2n+1 are,

k+ ≈√k2x − 16π2δ2

0(kx) (A–17a)

k2n+1 ≈√(2π + kx)2 − 16π2δ2

0(2π + kx) (A–17b)

To establish Eq.(3–37), we recall that the point k2n+1,lx is solution to the equation,√

k22n+1 − k2x

k22n+1 − (2π − kx)2=

l

2n + 1

Hence,

4πkx(1− 2r 2) + 4π2 = (1− r 2)�k , r =l

2n + 1

where k22n+1 = (2π + kx)2 − �k for �k given by Eq.(A–17b). Thus,

4πkx(1− 2r 2) + 4π2 = π2u(1− r 2)(2π + kx)4, u = R4(εc − 1)2

One can then look for a series solution kx = a0+a1u+a2u2+... This leads to Eq.(3–37).

115

APPENDIX BCOMPLEMENTS II

B.1 Estimation of ζ and ξ

Here the limit values ζb and ξb of the functions ζ and ξ defined in Eq. (4–22) are

estimated as h → hb along the resonance curve C . For ξb this is immediate. Indeed, in

the aforementioned limit, the field ratio η → 1 for a symmetric bound state and, therefore,

ξb = i4πδ0(k)

kz

∣∣∣∣∣k=kb

For ζb, the estimate follows from that wave numbers kb at which bound states exist are

close to the diffraction threshold 2π − kx when only one diffraction channel is open for

the fundamental harmonic, i.e., kx < k < 2π − kx .

Indeed, in the first order of δ0(k),

kb ≈ 2π − kx −8π2δ20(2π − kx)

2π − kx(B–1)

This proximity of the wavenumbers kb to the diffraction threshold 2π − kx allows one to

determine the leading terms in the coefficients a and b defined in Eq.(4–23), and hence

ζb. To proceed, the coefficients α2 = α(2k , 2kx) and β2 = β(2k , 2kx , h) of the matrix H2

are rewritten by separating explicitly the real and imaginary parts:

α2 + β2 = ψ+ + iSc , α2 − β2 = ψ− + iSs (B–2a)

where Sc and Ss are defined by the relations,

Sc = 16πδ0(k)∑mop,sh

cos2(hk shz ,m)

k shz ,m, Ss = 16πδ0(k)

∑mop,sh

sin2(hk shz ,m)

k shz ,m(B–2b)

and the index mop,sh indicates that the summations are to be taken over all open

diffraction channels for the second harmonic. Using the estimate (B–1), the functions ψ±

are found to obey the estimates,

ψ+ = 2 +O(δ0(kb)), ψ− = O(δ0(kb))

116

These expressions are then used to estimate ζ−1b = 2(a + b). In the first order of δ0(kb)

one infers that

ζb ≈ −1

4− 2πiδ0(k)

∑mop,sh

cos2(hk shz ,m)

k shz ,m

∣∣∣∣∣(h,k)=(hb,kb)

(B–3)

B.2 Complements on the Flux Analysis: Flux Conservation

For the nonlinear wave equation (4–1), the Poynting Theorem becomes,

1

8π

d

dt

[∫V

(εE 2 + B2 +

χ

3πE 3)dr

]= −

∫∂V

S · dn (B–4)

where V is a closed region, and ∂V is its boundary. The vector S = E×B is the Poynting

vector (for simplicity, it is assumed that ∂V lies in the vacuum so that µ = ε = 1, and

χ = 0 in a small neighborhood of ∂V ). In the case of a monochromatic incident wave,

the flux measured is the time-average of S over a time interval T → ∞. By averaging

Eq.(B–4), it then follows that,∫∂V

⟨S⟩ · dn = 0, ⟨S⟩ = 1

T

∫ T

0

S(t)dt, T → ∞

This is the flux conservation. In terms of the different harmonics of Eq.(4–3), the time

averaged Poynting vector becomes,

⟨S⟩ = − c2

2πωIm

(∞∑l=1

El∇E−l

l

)

Of interest is the flux of the Poynting vector across the rectangle depicted in Fig. 4-1(b).

By Bloch’s condition (4–4), the contributions to the flux from the faces L±1/2 : x = ±12

cancel out so that the flux measured is through the vertical faces L± = {(x , z)| − 12≤ x ≤

12, z → ±∞}. Note that the vanishing of the flux across the faces L±1/2 is a consequence

of the fact that the incident wave is uniformly extended over the whole x−axis. For

example, consider the normal incidence (kx = 0) with one diffraction channel open

for the incident radiation. Then the Poynting vector of the reflected and transmitted

fundamental harmonic is normal to the structure and, hence, carries no flux across

117

L±1/2. The second harmonic (l = 2) has three forward and three backward scattering

channels open, m = 0,±1, relative to the z−axis. The wave with m = 0 propagates

in the direction normal to the structure and does not contribute to the flux across L±1/2.

Since the incident wave has an infinite front along the x−axis, so do the scattered waves

with m = ±1. The waves with m = 1 and m = −1 carry opposite fluxes across each of

the faces L±1/2 as the corresponding wave vectors have the same z−components and

opposite x−components and, hence, the total flux vanishes. For a finite wave front (but

much larger than the structure period), the second harmonic would carry the energy flux

in all the directions parallel to the corresponding wave vectors in each open diffraction

channel.

If σl is as defined in Section 4.4, then the flux conservation implies that∑∞

l=1 σl = 1.

Therefore, in the perturbation theory used, i.e., when the system (4–5) is truncated to

Eqs.(4–9), the inequality σ1 + σ2 ≤ 1 must be verified to justify the validity of the theory.

The conversion ratio σ2 is given in Section 4.4. If only one diffraction channel is

open for the fundamental harmonic, then the ratio σ1 of the scattered and incident fluxes

of the fundamental harmonic reads,

σ1 = |1 + T0|2 + |R0|2

where T0 and R0 are the transmission and reflection coefficients which are obtained

from the far-field amplitude of E1 as,

E1 →

e ir·k + R0e

ir·k−, z → −∞

(1 + T0)eir·k, z → +∞

118

where k = kxe1+kze3 is the incident wave vector and k− = kxe1−kze3 is the wave vector

of the reflected fundamental harmonic. It then follows from Eqs.(4–7) and (4–18) thatR0 = i

2πδ0(k)

kz

[(E1+ + 2νE2+E 1+)e

ihkz + (E1− + 2νE2−E 1−)e−ihkz

]T0 = i

2πδ0(k)

kz

[(E1+ + 2νE2+E 1+)e

−ihkz + (E1− + 2νE2−E 1−)eihkz

]In the vicinity of a critical point (hb, kb), the coefficients R0 and T0 obey the estimate,

R0 ≈ T0 ≈ i4πδ0(kb)

kz ,bφE1+

(1 +

ν2|E1+|2

ζb

)After some algebraic manipulations, it is found that,

σ1 + σ2 = 1 +8πδ0(kb)

kz ,bν2|E1+|4

[Ab +

4πδ0(kb)

kz ,bφ2

(Re{1

ζb

}+ν2|E1+|2

|ζb|2

)]where Ab is the constant defined as,

Ab =

[2Sc |a + b + 1|2 − Im

{1

ζ

}](h,k)=(hb,kb)

and Sc = Im{α2 + β2} is introduced in Eqs.(B–2). Expressing a and b defined by (4–23)

via the coefficients α2 and β2 of the symmetric matrix H2, one also obtains

Sc = Im{

a + b

1 + a + b

}Since at the point (hb, kb) the value of ζ is ζb = (2(a + b))−1|(h,k)=(hb,kb), it follows that,

Ab =

[2Im

{a + b

1 + a + b

}|a + b + 1|2 − 2Im{a + b}

] ∣∣∣∣∣(h,k)=(hb,kb)

For general complex numbers a and b, the expression in square brackets is always zero.

Therefore Ab = 0, and,

σ1 + σ2 = 1 + 2

(4πδ0(kb)

kz ,bφν|E1+|2

)2(Re{1

ζb

}+ν2|E1+|2

|ζb|2

)(B–5)

119

By Eq. (B–3), Re{ζ−1b } ≈ −4. In Appendix B.3 it is proved that ν2|E1+|2 = O(φ2/3).

Consequently, near the critical point (hb, kb), the right hand summand in Eq.(B–5) is

negative so that σ1 + σ2 ≤ 1 as required.

B.3 Complements on the Amplitude E1

The amplitude of the field E1+ is a root of the cubic polynomial in Eq.(4–24) which

can be solved by Cardano’s method. Put Y = X + 23

(φν

)2 Re{ζξ}. For the new variable

Y , Eq. (4–24) assumes the standard form,

Y 3 + pY + q = 0 (B–6)

where,

p =φ4

3ν4(|ζξ|2 − 2Re{(ζξ)2}

), q =

2φ6

27ν6Re{ζξ}

(4Re{(ζξ)2} − 5|ζξ|2

)− φ2

ν4|ζ|4

As the amplitude E1 is uniquely defined by the system (4–5), it is therefore expected that

the cubic in Eq.(B–6) should have a unique real solution in order for the theory to be

consistent. The latter holds if and only if the discriminant

D3 =4

27p3 + q2

is nonnegative. To prove that D3 ≥ 0, note first that |ζξ|2 − 2Re{ζ2ξ2} > 0 in the vicinity

of a critical point (hb, kb). This follows from the estimates established in Appendix B.1.

Indeed, in the first order of δ0(kb),

|ζbξb|2 − 2Re{(ζbξb)2} ≈ 3π2

k2zδ20(k)

∣∣∣∣∣k=kb

> 0 (B–7)

Next, consider the complex number

ρ =4

27

[2Re{ζξ}

(2Re{(ζξ)2} − 5

2|ζξ|2

)+ i(|ζξ|2 − 2Re{(ζξ)2})

32

]

120

The positivity condition (B–7) ensures that the coefficient of the complex number i in the

expression of ρ is indeed real. After some algebraic manipulations, it can be shown that,

D3 =φ4

ν12

∣∣∣|ζ|2ν2 − φ4ρ∣∣∣2

Thus D3 ≥ 0 as required. The only real solution Y to Eq.(B–6) is then,

Y =3

√−q +

√D3

2+

3

√−q −

√D3

2

It then follows that,

|E1+| =|φ| 13ν

√τ+ + τ−, τ± = 3

√1

2

(ν2|ζ|2 − 1

2φ4Re{ρ} ±

∣∣∣|ζ|2ν2 − φ4ρ∣∣∣)− φ

43

3Re{ζξ}

(B–8)

provided τ+ + τ− ≥ 0. The latter condition imposes a limit on the validity of the

perturbation theory developed in the present study, i.e., the reduction of the system

(4–5) to (4–9) is justified if τ+ + τ− ≥ 0. This is to be expected because of the lack of

analyticity in χc of the solution to the nonlinear wave equation (4–1) that can only occur

at the critical points (hb, kb) at which bound states in the radiation continuum exist. As

one gets away from these critical points in the (h, k)-plane, the solution to the nonlinear

wave equation becomes analytic in χc , meaning that all the terms that were neglected in

finding the principal parts of the amplitudes must now also be taken into account to find

a solution befitting the series of Eq.(4–2). The shadowed region depicted in Fig. 4-2(b)

shows the region of the (h, k)-plane in which the condition τ+ + τ− ≥ 0 holds for the

first symmetric bound. The presented analysis of the efficiency of the second harmonic

generation is valid for any choice of the geometrical parameters, �h = h − hb and R,

and the physical parameters, εc and χc > 0, which satisfy the conditions (4–29a) and

τ+ + τ− ≥ 0.

121

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123

BIOGRAPHICAL SKETCH

Remy Friends Ndangali was born in 1980 in Rwanda. After the rwandan civil war,

he successively took refuge in the Democratic Republic of Congo, Kenya, and finally

Senegal. It is in the latter country that he graduated High School at Cours Secondaire

Sacre Coeur in 1999. He then studied Mathematical Sciences at Cheikh Anta Diop

University in Senegal, and graduated with an AEA in 2004. In the Fall of 2005, Remy

enrolled in the Graduate School of the University of Florida, and he earned his doctorate

in mathematics in 2011 under the guidance of Dr. Sergei Shabanov.

124